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The intestinal tract consists of multiple compartments with different functions, together ensuring digestion and uptake of nutrients, absorption of water and expulsion of remainders of food intake. The small intestine (SI) comprises villi that project into the gut lumen and crypts that invaginate into the mucosa. The large intestine (LI), comprising the caecum and colon, has a similar crypt architecture to the SI, but lacks luminal-projecting villi. In both regions, mitotic stem cell zones form at the bottom of crypts, with more differentiated epithelial cells positioned towards the gut lumen. The entire intestinal tube is lined with a single layer of epithelial cells that gets renewed every few days. This high turnover has been shown to be the consequence of dividing stem cells that compete neutrally for niche space at the base of each crypt1,2. After each stem cell division, one cell becomes displaced upwards along the crypt–lumen axis to differentiate into specialized cells such as goblet cells and enterocytes. Markers, including leucine-rich repeat-containing G-protein-coupled receptor 5 (LGR5), have been associated with stem cell activity3. However, such markers do not necessarily label all or exclusively cells that have the potential to form clones that persist over the long term4,5,6,7,8, which is considered here to be the defining property of effective stem cells. Using short-term intravital microscopy (IVM), we have previously shown that not every LGR5+ cell has the same survival potential and therefore the potential to act as a long-term effective stem cell9. Compared with LGR5+ cells positioned at the base of crypts, LGR5+ cells further away are more susceptible to passive displacement and subsequent loss and therefore have a lower chance to form clones over the long term9. However, the distance from the crypt base at which cells have a realistic chance to function as long-term stem cells—that is, the number of effective stem cells—is currently unknown owing to technical constraints in intravital microscopy that prohibit following of the same clone over several weeks. Moreover, it is unknown whether stem cells behave similarly in the SI and LI, as intravital microscopy has mainly been focused on the SI. Here we used intravital microscopy techniques that facilitate the monitoring of clones over multiple weeks in the SI and LI. By combining this analysis with 2D biophysical modelling of stem cell dynamics, we find that the numbers of effective stem cells in crypts differ between the SI and LI and that this is determined by the degree of Wnt-driven retrograde cell movement.

Characteristics of SI and LI crypts

To determine which cells act as effective stem cells in the SI and LI, we first looked at the total pool of LGR5+ cells at the crypt bases of Lgr5eGFP-Ires-creERT2 mice at both sites (Fig. 1a,b). The height of the LGR5+ zone is smaller in the LI than the SI, whereas the diameter of the crypt in the LI is larger than the SI (Fig. 1c,d). In both SI and LI crypts, LGR5+ cells are distributed over around four rows, with about five cells per row (Fig. 1b and Extended Data Fig. 1a). In both the SI and LI crypts, LGR5 expression is the highest for cells positioned at rows 0 and 1, and intermediate for cells positioned at rows 2 and 3, defined as centre and border LGR5+ cells, respectively (Fig. 1e,f). Cells positioned beyond row 3 express low levels of LGR5 (Fig. 1b). In both SI and LI crypts, the centre and border regions contain comparable numbers of LGR5+ cells (Fig. 1g), with a total of approximately 22–24 LGR5+ cells per crypt (Fig. 1g).

Fig. 1: The spatial organization and functional potential of LGR5+ cells are comparable in the SI and LI.
figure 1

a, Schematic of the crypt. b, Representative xy images of SI and LI crypts in Lgr5eGFP-Ires-creERT2 mice from n = 4 experiments, showing the relative positions of LGR5+ cells in the central (rows 0 and 1) and border regions (rows 2 and 3) of the stem cell niche. c,d, The height (c) and width (d) of the LGR5–GFP+ zone in SI and LI. For c and d, n = 50 (SI) and n = 48 (LI) crypts. e, f, The relative maximum intensity of LGR5–eGFP signal in the SI (e) and LI (f). n = 47, n = 62, n = 61 and n = 62 (e); and n = 66, n = 78, n = 77 and n = 84 (f) crypts for rows 0, 1, 2 and 3, respectively. g, The number of LGR5+ cells in the centre and border and the total number in SI and LI crypts. n = 50 (SI) and n = 48 (LI) crypts. h, Principal component analysis of RNA-seq data of LGR5high, LGR5medium and LGR5low cells (that is, centre, border and row >3 cells). The dots show the mean values. n = 3 mice. i,j, The organoid-forming efficiency of LGR5+ cells with high, medium and low intensity isolated from the SI (i) and LI (j). The dots show the percentage of cells that formed organoids in a BME drop. From left to right, n = 20, n = 20, n = 39, n = 38, n = 19 and n = 24 (i); and n = 13, n = 12, n = 32, n = 34, n = 19 and n = 24 (j) BME drops from n = 3 experiments in n = 3 mice. k, Confocal images of LGR5–eGFP cells in S phase (4 h EdU) and mitosis (phosphorylated histone H3 (PH3), arrow heads) in SI and LI crypts (dotted outline). l,m, EdU+ (l) and PH3+ (m) LGR5+ cells as a percentage of the total LGR5+ pool in the SI (n = 33 crypts (l); n = 3 image fields (m)) and LI (n = 22 crypts (l); n = 3 image fields (m)). Data are mean ± s.e.m. For c, d, l and m, significance was determined using two-sided Mann–Whitney U-tests. For b and k, scale bars, 20 μm.

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To test for potential differences in transcriptional level or colony-forming ability, we isolated LGR5+ cells from the centre, border and the region beyond the border (row >3) from both SI and LI crypts on the basis of their LGR5 expression using flow cytometry (Extended Data Fig. 1b). As expected, we observed differences in the transcriptional profile between centre and border LGR5+ cells within SI and LI crypts and more profound differences between the two intestinal sites (Fig. 1h and Extended Data Fig. 1c,d). Similar results were found when we performed staining for other (stem cell) markers and Wnt targets (Ascl2, Smoc2, Axin2, Cd44, Cycd1, Efnb2 and Efnb3) (Extended Data Fig. 1e–g). Importantly, regardless of any molecular differences, centre and border LGR5+ cells showed similar potential to form organoids in growth-factor rich medium (Fig. 1i,j). These data suggest that LGR5+ cells, regardless of their position within the crypt, have the potential to regain the full clonogenic ability when exposed to factors from the stem cell niche.

In addition to the number of LGR5+ cells and their intrinsic molecular and functional potential, the cell division rate and the position of proliferating cells are important for the dynamics of stem cell competition as, together, they determine the rate at which stem cells are replaced in the niche. We compared the presence of proliferating cells in the SI and LI crypts by quantifying the number of cells in S phase (measuring short-term 5-ethynyl-2-deoxyuridine (EdU) incorporation) and the number of mitotic cells (positive for phosphorylated histone H3) (Fig. 1k–m). Our analyses revealed similar numbers of proliferating cells in the central crypt regions of the SI and LI (Fig. 1l,m). Moreover, we found that more cells in the border region of SI crypts were proliferating compared with those in the LI (Fig. 1l,m), resulting in a slightly higher fraction of proliferative cells in the SI overall. Notably, the proliferative zone above the LGR5+ zone, known in the SI as the transit-amplifying zone, was significantly less pronounced in the colonic epithelium compared with the SI when assessed by BrdU incorporation (Extended Data Fig. 1h). Together, these results indicate that the spatial organization of LGR5+ cells within crypts and their functional potential are overall comparable between the different intestinal compartments.

Effective stem cell numbers in the SI and LI

As the crypt architecture, including the distribution of cells with the same potential to regain the full clonogenic ability, was found to be similar, we next tested whether SI and LI crypts also contained similar numbers of effective stem cells. For this, we used our repetitive intravital microscopy techniques that enabled us to trace the progenies of LGR5+ cells at different positions within the crypt over several weeks. The vasculature in large overview images was used as a landmark to find the same region, while images with higher resolution and magnification of the patchy expression of GFP and sporadic distribution of confetti-coloured crypts were used to validate the successful tracing of regions of interest (Extended Data Fig. 2a,b). This approach enabled us to retrace the same crypts over multiple imaging sessions and enabled us to study the fate of individual LGR5+ cells over periods of months (Fig. 2a,b).

Fig. 2: Different numbers of effective stem cells in the SI and LI due to retrograde movement.
figure 2

a, Schematic of the experimental set-up. b, Representative overview images 48 h and 8 weeks after tracing in the SI (left) and LI (right) from 6 independent experiments. The dotted yellow and white circles represent retraced crypts. A labelled clone can either be lost (loser, dashed yellow circle) or retained (winner, dashed white circle). c, Clone retention from different starting positions in the SI (n = 267 clones in n = 6 mice) and LI (n = 294 clones in n = 6 mice). d, Normalized retention probability at 8 weeks as predicted by the model (lines with 95% CI) and experimental data (dots) in the SI (n = 267 clones in n = 6 mice) and LI (n = 294 clones in n = 6 mice). e, Model sketch: the crypt is abstracted as a cylinder coupled to a hemispheric region (i). kd is the upward movement rate due to cell division (ii and iii) and kr is the random cell relocation rate, including retrograde movements (iv). f, Example of a border-starting clone at day 1 that moved to the centre at day 4 (retrograde movement) from 7 independent experiments. g, The percentage of border-starting clones present at the centre (retrograde movement) on day 3 in the SI (n = 59 clones in n = 7 mice) and LI (n = 109 clones in n = 4 mice). Data are mean ± s.e.m. Significance was determined using two-sided Mann–Whitney U-tests. Scale bars, 50 µm (b), 20 µm (f).

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Using the multiday imaging approach, we recorded the locations of clones in crypt bases 48 h after induction and traced their fate 8 weeks later in Lgr5eGFP-Ires-creERT2;R26R-Confetti mice (Fig. 2a,b). Cells that gave rise to clones that persisted over the long-term were designated as effective stem cells (Fig. 2b (winner)). Quantification of clone retention (that is, the percentage of clones that remained present in the LGR5+ zone) revealed that centre-derived clones were more likely to persist than border-derived clones, as expected on the basis of previous studies9. Notably, in the LI, no border-derived clones remained in the stem cell niche 8 weeks after the onset of tracing, whereas, in the SI, around 15% of border-derived clones continued to persist (Fig. 2c). Further dissection of different starting positions within the crypt bases revealed a gradient of positional advantage and clone retention probability, decreasing from centre to border (Fig. 2c,d), which was substantially steeper in the LI compared with the SI (Fig. 2d). These results indicate that, in contrast to the SI, LGR5+ cells in the border of LI crypts do not function as effective stem cells. Together, we conclude that, during homeostasis, SI crypts contain more effective stem cells than LI crypts.

Modelling suggests retrograde movements

To better understand why LI border LGR5+ cells do not function as effective stem cells while they have the intrinsic molecular and functional potential to do so, we turned to quantitative modelling. Previous studies have sought to model the neutral drift dynamics of clones around the crypt circumference using a minimal one-dimensional scheme1 in which the effective stem cell number was linked to LGR5+ expression2 or fit using a continuous labelling strategy10. However, to understand how this effective stem cell number arises in the first place11, we explicitly modelled the two-dimensional organization and biophysical cellular dynamics of the intestinal crypt (Supplementary Notes 1.1 and 2.1). In particular, we modelled individual crypts as regular two-dimensional cylindrical grids (denoting rows 0, 1 and so on as the cell position along the crypt–villus axis, with 5 cells per row, as measured in Extended Data Fig. 1a). In this model, cells undergo two core processes (sketched in Fig. 2e): (1) cell division at rate kd (with random division orientation), leading to the upward transfer of cells along the crypt–villus axis (that is, anterograde movement); and (2) random cell relocation at constant rate kr, leading to the exchange of neighbouring cells either within or between adjacent rows, allowing for additional directed onward or retrograde movement towards the crypt base. In the absence of retrograde movement (termed the deterministic conveyor belt model), cells at row 0 would systematically ‘win’ the competition over cells at rows >0, whereas non-zero values of kr (the stochastic conveyor belt model) allow for cells at higher rows to participate in the competition, and function as effective stem cells. Crucially, this model makes the generic prediction that the probability of long-term clone retention should decrease with the distance from the bottom of the crypt as a Gaussian-like distribution, specifying a well-defined effective stem cell number that depends on only the dimensionless ratio kr/kd (that is, the ratio of relocation and division rates; Supplementary Note 1.2). Thus, this modelling suggests that differences in retrograde movement (that is, kr) could provide a mechanistic explanation of the differential fate of the border LGR5+ cells in the SI and LI.

Imaging retrograde movement intravitally

To test whether the difference in the ability of border LGR5+ cells to function as effective stem cells is caused by differential retrograde movement in SI and LI crypts, we performed short-term multiday intravital microscopy in living Lgr5eGFP-Ires-creERT2;R26R-Confetti or Lgr5eGFP-Ires-creERT2;R26-LSL-tdTomato mice. Injection of a low dose of tamoxifen resulted in Cre-mediated recombination and activation of one of the Confetti colours or tdTomato, respectively, in individual LGR5-expressing cells. Using intravital microscopy on consecutive days, we monitored cell movement and clonal evolution of cells originally located at central or border regions over time in both the SI and LI (Fig. 2f). Positions and clone persistence were determined and quantified (Fig. 2f and Extended Data Figs. 2c,d and  3–6). As expected qualitatively from the model, retrograde movement of labelled cells was observed in the crypts of the SI, but was nearly absent in LI crypts (Fig. 2g). Consistent with the difference in retrograde movement, we confirmed the modelling predictions that clones that started from positions away from the crypt base (at levels 2 and 3) in LI crypts are lost faster than their counterparts in SI crypts (Fig. 2c,d and Extended Data Fig. 2c,d).

Wnt stimulates migration of LGR5+ cells

Next, we aimed to determine the factors that drive retrograde cell movement. Recently, Wnt signalling has been shown to induce migration of intestinal stem cells in Drosophila12. Wnt has been implicated as one of the major niche factors to induce stem cell potential, and is produced by Paneth cells (PCs) and pericryptal stromal cells in SI crypts13. By contrast, LI crypts do not contain PCs13. To test whether PC-produced Wnt has the potential to induce migration of LGR5+ cells, we isolated LGR5+ cells by flow cytometry. Next, we seeded these cells onto ultra-low-attachment round-bottom plates and imaged their migration using confocal microscopy as previously described14 (Supplementary Video 1). Interestingly, we observed that the mean square displacement of migratory cells was enhanced when cells were exposed to PCs (Fig. 3a,b), an effect caused by both enhanced speed and directional persistence (Extended Data Fig. 7). Importantly, this migration enhancement was abolished when the secretion of Wnt was inhibited by a porcupine inhibitor (IWP2) (Fig. 3a,b). These results suggest that PC-secreted Wnt can promote the migration of LGR5+ cells. Indeed, when we directly stimulated LGR5+ cells with exogenous Wnt, migration was enhanced even further (Fig. 3a,b and Extended Data Fig. 7).

Fig. 3: Wnt promotes LGR5+ cell migration.
figure 3

a, Normalized migration tracks of single LGR5+ cells isolated from LGR5-EGFP-ires-creERT2;R26R-confetti organoids in control medium; medium supplemented with WNT3A; co-culture with PCs; or co-culture PC with Wnt inhibitor (IWP2) in Matrigel. n = 150 random tracks of LGR5+ cells from n = 2 organoid lines, n = 3 biological replicates. b, Mean square displacement calculated as a function of time. n = 150 cells, n = 3 biological replicates. Data are mean ± s.e.m. c, Schematic of the experimental set-up for analysing cell movement on decellularized intestinal scaffolds. d, The motility of LGR5+ cells was determined along the crypt–villus axis (motility in the z axis) and along the lateral axis (motility in xy axes). n = 150–200 cell tracks from n = 3 decellularized intestines for each control- (vehicle) and LGK974-treated group. Data are mean ± 95% confidence interval. e, The percentage of border-starting clones present in the crypt centre (retrograde movement) on day 3. n = 59, n = 109 and n = 75 clones in n = 7, n = 4 and n = 4 mice for the SI and LI and the SI after LGK974 treatment, respectively. Data are mean ± s.e.m. Significance was determined using two-sided Mann–Whitney U-tests. f, Quantification of retention within the LGR5+ zone of centre- or border-starting clones in the crypts of control (solid lines) and LGK974-treated (dashed lines) mice as followed by IVM. n = 75 clones in n = 4 mice. g, The normalized retention probability in the LGR5+ zone in LGK974-treated SI (n = 75 clones in n = 4 mice; data reanalysed from ref. 16) predicted by model (solid lines, mean ± 95% CI) and experimental data (dots). h, The probability of a clone starting either in the niche centre (left) or border (right) to be present in the centre, border or to be lost from the LGR5+ zone over time in the SI of LGK974-treated mice, comparing data (left bar) and theory (right bar). n = 75 clones in n = 4 mice (Supplementary Note 2.2.3). For c, scale bar, 50 μm.

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To further validate the induction of retrograde movement by Wnt signalling, we decellularized the intestines from control and porcupine-inhibitor-treated mice (Fig. 3c). Such decellularized extracellular matrix scaffolds retain the original intestinal tissue architecture and multiple ECM bound factors that can provide positional cues guiding cell movement in vivo15 (Fig. 3c). Next, we seeded single LGR5+ cells on top of these scaffolds and observed using live microscopy that individual cells displayed highest motility along the z axis in crypts, an effect that was absent in villi (Fig. 3d and Supplementary Video 2). Moreover, cell motility along the z axis was decreased specifically in the crypts of those scaffolds that were prepared from mice treated with the porcupine inhibitor LGK974 (Fig. 3d).

Finally, we tested whether inhibition of Wnt release also blocks the retrograde movement that we observe in vivo. Previously, we have shown that reduced Wnt secretion after treatment with a low dose of LGK974 results in a lower number of effective stem cells16. By reanalysing16 our intravital microscopy data, we indeed found that this change was correlated with a near abolition of retrograde cell movement (Fig. 3e) as well as decreased clone retention (Fig. 3f), whereas the cell proliferation rate was affected minimally by LGK974 treatment (Extended Data Fig. 8a,b). When combined, in addition to its classical role as a promoter of self-renewal in the base of crypts, our data suggest that Wnt stimulates cell migration and mediates retrograde cell movement.

Retrograde movement affects stemness

Our modelling and experiments suggest that Wnt-dependent retrograde movements enable border LGR5+ cells to function as stem cells (that is, to function as an effective stem cell). However, the extent of this effect and the long-term consequence of differences in retrograde movement (that is, kr) on effective stem cell number in the SI and LI crypts remained to be investigated quantitatively. The predicted Gaussian dependency of the probability of clone persistence as a function of initial position along the crypt–villus axis enabled us to define the number of effective stem cells in crypts (Fig. 2d) as the number of cells of which the lineage has a significant probability to persist over the long term (set above 5% for a given row, comprising 2σ of the fluctuations, where σ2 ∝ kr/kd; Supplementary Note 1.2).

Consistent with this simple prediction, which did not depend on the decided implementation of retrograde movements in the model (Supplementary Note 1.4), the clonal persistence at 8 weeks after induction as a function of starting position was well fitted by a Gaussian function in both the SI and LI. From this fit (Fig. 2d and Supplementary Note 2.2), we obtained: kr/kd = 0.25 in the LI (95% confidence interval (CI) = 0.05–0.55) and kr/kd = 2 in the SI (95% CI = 0.5–4.5).

We next estimated the consequence of differential kr on the number of effective stem cells, Ns, using the predicted relationship (Supplementary Note 1.3):

$${N}_{{\rm{s}}}\approx {N}_{{\rm{g}}}\left(1+2\sqrt{\frac{{k}_{{\rm{r}}}}{{k}_{{\rm{d}}}}}\right)\,$$

where Ng = 5 is the number of cells per row (Extended Data Fig. 1a). This led to a figure of NsSI ≈ 19 effective stem cells per crypt in the SI (accounting for nearly all LGR5+ cells) and only around NsLI ≈ 10 effective stem cells per crypt (translating to the first two rows) in the LI.

To validate experimentally the consequence of retrograde cell movement on the number of effective stem cells, we compared the evolution of clone retention as a function of starting position to predictions from numerical simulation and analytical theory, using the same values of kr/kd inferred above. For the short-term dynamics, the proliferation rate kd also enters in the prediction, setting the overall time scale of the dynamics, with its inferred value broadly consistent with EdU data despite some variations between best-fit value between conditions, which could be related to short-term biases during lineage tracing (Supplementary Note 1 and Extended Data Fig. 9a–g). These results provided a good quantitative agreement at every time point for both the SI and LI (Extended Data Fig. 9h,i). Furthermore, the model and data showed an excellent quantitative agreement (the statistics are provided in Supplementary Note 2) for the retrograde movement of clones, indicating that differences in both short- and long-term dynamics in the SI and LI can be captured by our model (Extended Data Fig. 9j–m). Thus, the analysis of the short-term dynamics corroborates the findings of long-term lineage data and the values of kr/kd extracted above, confirming experimentally that the retrograde movement determines the effective stem cell number in the SI and LI crypts.

Next, we perturbed retrograde movement by inhibiting Wnt signalling using the porcupine inhibitor LGK974. Fitting the model to the dynamics of clone retention as a function of starting position confirmed that LGK974 treatment decreased kr/kd to 0.4 (95% CI = 0–1), resembling the value found in the LI during homeostasis. As well as providing a good fit of the clone retention dynamics (Fig. 3g), this ratio also predicted the rate of movement of clones between centre and border compartments (Fig. 3h and Supplementary Note 2.2).

Finally, we reasoned that a large retrograde movement rate kr in the SI should also have a consequence for the spatial dispersion of clones along the crypt–villus axis. To test this, we performed short-term tracing, for which we reconstructed clones along the crypt–villus axis, both in the centre-border and beyond-border compartments. Importantly, we found that the probability of clone fragmentation was much higher in the SI than in the LI (Extended Data Fig. 10), and that the magnitude of the difference was well predicted by our 2D spatial model with the same parameters kr/kd extracted above, providing an independent confirmation for the predictions of the model. Together, these experiments further strengthen our conclusions that different short-term retrograde movement can explain differences in the long-term retention probability of crypt cells, and thereby dictate different effective stem cell numbers in SI and LI crypts.

Retrograde movement alters monoclonality

To examine the consequences of different effective stem cell numbers in SI and LI crypts, despite showing similar crypt characteristics and near-equal number of LGR5+ cells, we next compared the time that it takes for one clone to outcompete all others in a given crypt, that is, the time that it takes to reach crypt monoclonality. For this, we used whole-mount preparations of Lgr5eGFP-Ires-creERT2;R26R-Confetti mice that were euthanized at different time points after onset of lineage tracing (Fig. 4a). As predicted by our model, we observed differences in clonal expansion over time within the LGR5+ zone of SI and LI crypts, with faster evolution (that is, growth) of LI clones compared with SI clones (Fig. 4b). Monoclonality was reached faster in the LI than in the SI—whereas only around 30% of SI crypts were monoclonal at 6 weeks after onset of tracing, the vast majority of LI crypts was already monoclonal at that time (Fig. 4a,b). Notably, this evolution was predicted quantitatively by our model (Fig. 4c) with the different estimated ratios of kr/kd for each region (while the geometry—that is, the number of cells competing neutrally in a given row—and average cell division rate were equal). Moreover, our model prediction showed good agreement with the experimental long-term clonal evolution data after porcupine inhibitor treatment (Fig. 4d and Supplementary Note 2.4). Together, this argues that the process of monoclonal drift in intestinal crypts is faster in the LI than the SI, and can be enhanced by lowering the number of effective stem cells owing to reduced retrograde movement after porcupine inhibition.

Fig. 4: The consequences of retrograde movement.
figure 4

a, Representative maximum projections of crypt bottoms 8 weeks after induction in the SI and LI. n = 5 independent experiments. b, Heat map showing the frequency of clone sizes at different time points in the SI and LI. c, Monoclonal crypts in the SI and LI over time predicted by model (solid line) and real data in the SI and LI. Data are mean ± s.d. For b and c, n = 444, 375, 218, 152 and 151 clones in n = 3, 4, 3, 5 and 7 mice (SI); and n = 304, 465, 236, 164 and 531 clones in n = 3, 4, 3, 5 and 8 mice (LI) for 1, 2, 4, 6 and 8 weeks, respectively. d, Monoclonal crypts in the SI of control and LGK974-treated mice (n = 3, 3, 4, 4, 3 and 3 (control) and n = 2, 4, 3, 5, 3 and 3 (LGK974) mice for days 4, 7, 10, 21, 30 and 50, respectively; n = 200 clones per mouse) over time predicted by model (solid line) and real data reanalysed from ref. 16. Data are mean ± s.e.m. (Supplementary Note 2.2.3). e, Schematic of DT treatment. Targeted ablation of LGR5+ cells by two DT injections in 19 Lgr5DTR:EGFP mice. f, Representative images of untreated and DT-treated mice at different time points after ablation. n = 3 independent experiments. g, Heat map of the percentage of crypts recovered in the centre and border regions in the SI and LI. n = 5, 3, 4 and 3 mice for days 2, 4, 7 and 15, respectively. Note the faster recovery kinetics of SI crypts (white dashed triangles). h, The best numerical fit for the position of the lowest LGR5+ cell using the biophysical model (left), predicting faster dynamics in the presence of a larger amount of retrograde movements, and experimental data (right). n = 2, 3, 5, 3, 4 and 3 mice for control, day 1, 2, 4, 7 and 15, respectively. Scale bars, 200 μm (a) and 100 μm (f).

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Retrograde movement affects regeneration

Movement of cells from the transit-amplifying zone into the stem cell zone has been shown to be important for the regeneration of the stem cell pool following their loss17. Thus, to test whether distinct retrograde movement in SI and LI crypts also affects regeneration of the stem cell pool, we traced the recovery of the SI and LI LGR5+ cell compartment after targeted ablation of LGR5+ cells using diphtheria toxin (DT) injection in mice in which the human DT receptor (DTR) fused to eGFP was knocked into the Lgr5 locus (Lgr5DTR:eGFP)17. We measured the recovery of LGR5–DTR–GFP cells at various cell positions in the crypt (Fig. 4e–g). We first performed this experiment in silico using our mathematical framework (Fig. 4h (left)). Owing to differences in the retrograde movement, our framework predicted that the recovery of the stem cell pool in the SI should be faster than in the LI, especially regarding the stem cells in the centre of the niche (position 0 and 1). Notably, we indeed observed such distinct behaviour in the experiment (Fig. 4h (right)). Although the recovery of centre LGR5+ cells was nearly complete after 4 days in the SI, a large proportion of the LI crypts still lacked a centre LGR5+ cell after 7 days (Fig. 4h (right) and 4g (white dotted area)). Combined, these data suggest that the retrograde movement supports fast regeneration of the LGR5+ cell pool.

Discussion

Stem cells are defined by their potential to self-renew and give rise to differentiated progeny. Traditionally, stem cell potential has been thought to be an intrinsic property that is induced by cues such as Wnt, and characterized by signature expression of molecular markers such as Lgr5. Although LGR5+ cells in intestinal crypts have the intrinsic ability to give rise to all differentiated cell types and to form in vitro organoid cultures, LGR5 expression does not necessarily overlap with the ensemble of cells showing functional stem cell activity4,5,6,7,8. Here we have shown that, in addition to the passive rearrangements of cells after proliferation (that is, cells that are displaced by others), Wnt-driven active retrograde movement (that is, cells that move by themselves in the opposite direction of anterograde movement) determines the number of effective stem cells in intestinal crypts. Although LGR5+ cells at the border of the LI crypt base (rows 2 and 3) are exposed to the stem-cell-inducing niche and have all of the properties to function as stem cells, we find that, owing to lack of retrograde movement, these cells are not able to form long-term clones and act as effective stem cells as they get continuously displaced by the progeny of LGR5+ cells located at lower rows. By contrast, owing to the presence of retrograde movement in SI crypts, cells at the border of the crypt base, possibly including those outside the LGR5+ zone, can take turns to visit the crypt centre, and therefore have a chance to function as long-term effective stem cells. Thus, we conclude that stem cell identity is neither cell intrinsic nor marked by distinct molecular signatures but, rather, is defined by the potential of cells to be present in or enter a location that both induces self-renewal and prevents displacement.

Our experimental data and mathematical modelling demonstrate the importance of random cell relocation and retrograde movement in governing stem cell identity, and show surprising differences in the rate and dynamics of monoclonal conversion in the SI versus LI, despite strong similarities in crypt characteristics. Our model identifies a single dimensionless parameter, that is, the ratio between cell repositioning rate and division rate (kr/kd), which sets the spatial extent of the effective stem cell region, defined as the length along the axis of the crypt at which cells have a significant probability to contribute to long-term renewal. In the LI, kr and, therefore, retrograde movement is so minimal that self-renewal potential is largely limited to cells positioned at row 0 (that is, approximately 5–10 stem cells compete laterally and neutrally), whereas more complex dynamics emerge in the SI due to retrograde movements, with neutral competition between cells of the same row1,2 as well as biased competition between cells of different rows (stochastic conveyor belt).

Importantly, the differential rate of cell repositioning and retrograde movement was sufficient to explain the differential dynamics underlying regeneration of the stem cell pool, competition for niche space in the SI and LI, as well as the faster clonal evolution and shorter fixation times in the LI compared to the SI. This underlines the importance of a better understanding of the molecular and cellular bases of retrograde movements. Interestingly, we observed less retrograde movement after porcupine inhibition, whereas proliferation was unaltered. This was consistent with complementary experiments from in vitro and decellularized migration assays, indicating a role for Wnt signalling in driving active migratory movements even in regions beyond the crypt base with low levels of Wnt.

To induce self-renewal in the Wnt-rich region, the pericryptal stroma is a sufficient Wnt source in the absence of PCs18,19. However, for retrograde movement, this source is insufficient as this movement is not present in the LI. Seemingly, for movements in regions beyond the crypt base, other Wnt sources, such as PCs in the SI, are required. Although the low levels of Wnt outside the crypt base may be insufficient to induce self-renewal properties, our data suggest that these levels are high enough to induce migration enabling cells to enter Wnt-rich regions. As Wnt is known to regulate numerous other processes, including induction of self-renewal, it remains difficult to directly provide causality between stem cell function and retrograde movement. Nevertheless, our results suggest that retrograde movements in the SI are not only a result of random repositioning during cell division, but represent a new level of active stem cell regulation that can be experimentally and pharmacologically manipulated. In this regard, it is also interesting that Wnt-antagonists are increased during ageing20 and early adenoma formation21. Given the importance of homeostatic stem cell dynamics in predicting the response of the cellular compartments to non-neutral mutations, the retrograde movement shown here may contribute to susceptibility to pathological conditions such as tumour initiation along the intestinal tract.

Methods

Mice and treatment

All animal experiments were carried out in accordance under the guidelines and approval of the animal welfare committee of the Netherlands Cancer Institute and Hubrecht Institute (KNAW). The following mouse strains were used: Lgr5eGFP-Ires-creERT23; R26-LSL-tdTomato22 and R26-confetti2. For lineage tracing experiments, Lgr5eGFP-Ires-creERT2;R26-Confetti and Lgr5eGFP-ires-creERT2;LSL-tdTomato mice were used. Lgr5eGFP-Ires-creERT2 were used to isolate LGR5+ cells and C57/B6 mice were used to decellularize intestines for the decellularized intestine experiment. LGR5-DTR-GFP17 mice were used for the LGR5+ cell ablation experiments. Genentech provided the LGR5-DTR-GFP mice through their MTA program. Mice (mixed or C57/B6 background) were housed under standard laboratory conditions in SPF cages in an animal room at constant temperature (19–23 °C) and regulated humidity under a 12 h–12 h light–dark cycle and received standard laboratory chow and water ad libitum. Male and female mice between 8 and 50 weeks of age were used for static lineage tracing and IVM experiments. For whole-mount imaging and imaging of isolated crypts, RNA-seq, colony-forming assay, decellularized intestine, LGR5+ cell ablation and clone dispersion experiments, intestines from 8–50 week-old male and female mice were used. Strains were bred in house. LGR5eGFP-Ires-creERT2;R26-Confetti received 1 mg tamoxifen (Sigma-Aldrich) for IVM lineage tracing experiments and 5 mg tamoxifen (Sigma-Aldrich) for static lineage-tracing experiments. For long-term (intravital) tracing, small and large intestines were taken from the same mouse. This was not possible for short-term intravital microscopy experiments, as only either the small or large intestine fitted behind the abdominal imaging window (AIW). For the decellularized intestine experiment, the porcupine inhibitor LGK974 was administered at a concentration of 5 mg kg−1 BID (oral) in a vehicle of 0.5% Tween-80/0.5% methylcellulose. Mice of the same litter were randomly assigned to each condition regardless of the sex. Different conditions were imaged and analysed in a random manner to minimize potential cofounding factors. No statistical methods were used to predetermine sample size estimates.

Surgery

All surgical procedures were performed under ~2% isoflurane (v/v) inhalation anaesthesia. Before and 8–12 h after surgery, mice were treated with a dose of buprenorphine (subcutaneous, 100 μg kg−1 mouse, Temgesic; BD Pharmaceutical System). Rimadyl (64 μg ml−1, Carprofen; Zoetis) was given in drinking water for 3 days after the surgery. For short-term intravital microscopy, an AIW was placed as described previously23. In brief, the left lateral flank of the mouse was shaved and disinfected. An incision was made through the skin and peritoneum of the mouse and a purse string suture was placed along the edge of the wound. The ileum (SI) or caecum (LI) was exposed and a disinfected AIW (>1 h in 70% (v/v) ethanol) was placed on top. In the case of the ileum, the mesenterium was fixed to the cover glass using Cyanoacrylate Glue (Pattex) and CyGel (BioStatus) was added on top to prevent liquid accumulation. In the case of the caecum, it was fixed to the titanium ring of the AIW using Cyanoacrylate Glue (Pattex). When these substances were dry, the intestines and AIW were placed back into the abdominal cavity and the skin and abdominal wall were placed into the groove of the AIW. Subsequently, the suture was tightened. After surgery, the mice were closely monitored daily for reactivity, behaviour, appearance and defecation. For repetitive long-term imaging, parts of the intestine that were imaged were exteriorized through a midline abdominal incision. Tissue hydration was maintained by creating a wet chamber, covering the mice with parafilm and the exposed tissue with PBS-drenched gauze. After the imaging session imaged tissue was placed back in the abdomen and the abdomen was closed using vicryl absorbable sutures (GMED Healthcare).

Intravital microscopy

For every imaging session, mice were sedated using isoflurane inhalation anaesthesia (∼1.5% isoflurane/O2 mixture) and placed into a custom-designed imaging box. For short-term imaging, mice were imaged once a day for a maximum of 3 h. For long-term imaging, mice were imaged 2–3 days after label induction and 8 weeks thereafter. z-Stacks and overview images were recorded using the Navigator function from Leica. The patchy expression pattern of the Lgr5 knockin allele, in combination with specific landmarks such as blood vessels, enabled repeated identification of imaged areas over consecutive days. After imaging, only if necessary, the acquired images were corrected for bleed through, cropped, smoothened, rotated and contrasted linearly in Fiji (v.2.3.0) (https://imagej.nih.gov/ij/).

Whole-mount preparation

For whole-mount imaging, intestines were collected and the lumen was flushed with ice-cold PBSO. The tissues were opened longitudinally and for the ileum, villi were removed from the luminal surface using a cover glass. The tissues were washed in ice-cold PBSO and fixed for 30 min in 4% formaldehyde solution (w/v) (Klinipath)) or periodate-lysine-4% paraformaldehyde (PLP) overnight at 4 °C (ref. 24). For antibody labelling, the tissues were permeabilized in 0.8% Triton X-100 in PBS containing 3% BSA. Subsequently, stretches of ~2 cm of fixed tissue, were mounted between 2 coverslips and embedded in Vectashield HardSet Antifade Mounting Medium (Vector Laboratories). Crypts were imaged from the bottom using the same equipment and settings as for intravital microscopy described below. For storage, PLP-fixed tissues were incubated in sucrose for >6 h and frozen in OCT at −80 °C.

Crypt isolation

For crypt isolation, intestines were collected and the lumen was flushed with ice-cold PBS. Tissue was opened longitudinally, villi were removed from the luminal surface of distal ileum. Parts of approximately 3 cm of ileum and intact but opened caecum were incubated with 30 mM in EDTA in HBSS at room temperature for 20 min. After vigorously shaking, the release of the epithelium from the mesenchyme was checked using a microscope. Suspensions were filtered (100 μm) before centrifuging (5 min at 4 °C, 88 rcf). Pellets containing isolated crypts were washed with cold PBS, fixed in 4% PFA (30 min at room temperature), permeabilized in 1% Triton X-100 (45 min at room temperature), blocked in blocking buffer for 30 min at room temperature (1% BSA, 3% horse serum, 0.2% Triton X-100 in PBS) before antibody labelling.

Cell proliferation and antibody labelling

To label cells in S phase, 1 mg of EdU (200 μl in PBS) or 2 mg bromodeoxyuridine (BrdU, 200 μl in PBS) was injected intraperitoneally 2 h or 4 h before euthanasia (indicated in the figure legends). Tissues were processed for whole-mount analysis or crypt isolation as described above. The Click-it staining reaction was performed according to the manufacturer’s protocol (Click-it EdU, Thermo Fisher Scientific/Invitrogen). For labelling of BrdU incorporation, crypts were incubated in 2 N HCl at 37 °C for 15 min to denature the DNA followed by 15 min in 0.1 M sodium borate for neutralization before incubation with anti-BrdU antibodies (1:50, Abcam, 6326) and anti-GFP antibodies (1:200, Abcam, 6673) overnight. To label cells during mitosis, anti-phosphorylated histone H3 antibodies were used (1:200, Millipore, 06-570). To visualize ephrin B2 and B3, intestines were stained with anti-ephrin-B2 antibodies (1:100, R&D Systems, AF467) and anti-ephrin-B3 antibodies (1:100 R&D Systems, AF432). Stainings were finalized by incubation with Alexa Fluor secondary antibodies, donkey anti-goat IgG Alexa Fluor 488 (A-11055), donkey anti-rabbit IgG (H+L) Alexa Fluor 568 (A-10042) and chicken anti-rabbit IgG (H+L) Alexa Fluor 647 (A-21443) (1:200 Invitrogen) combined with DAPI followed by mounting in antifading mounting medium (Vectashield, Vector laboratories). To visualize Wnt target gene expression, formalin-fixed intestinal tissue was incubated with the following Primary antibodies as previously described21: CD44 (1:50 BD Biosciences, 550538), cyclin D1 (1:50, Dako, M3635).

RNA in situ hybridization

In situ hybridization for Lgr5 (312178), Smoc2 (318548), Ascl2 (412218) and Axin2 (400338) mRNA (all from Advanced Cell Diagnostics) was performed using RNAscope 2.5 LS Reagent Kit–BROWN (Advanced Cell Diagnostics) on a BOND RX autostainer (Leica) according to the manufacturer’s instructions.

Microscopy equipment and settings

Tissues were imaged with an inverted Leica TCS SP8 confocal microscope. All images were collected in 12 bit with ×25 water immersion objective (HC FLUOTAR L N.A. 0.95 W VISIR 0.17 FWD 2.4 mm).

Sample preparation for RNA-seq and colony-formation assay

Small and large intestinal crypts were isolated as previously described25. To obtain single cells, crypts were treated with TrypLE (Life Technology) supplemented with 30 µg ml−1 DNase I (Sigma-Aldrich) and Y-27632 (10 µM) (incubation at 37 °C for 30 min). Dissociated cells were filtrated through a 100 µm cell strainer (Greiner Bio-One), and cells were resuspended and incubated with antibodies for 30 min on ice in 1 ml PBS containing FCS 5%, EDTA 5 mM (Accugene, Lonza), B27 2% (Thermo Fisher Scientific, 17504-044), N-acetylcysteine 1.25 mM (Sigma-Aldrich, A9165), mEGF 50 ng ml−1 (Peprotech, 315-09), noggin and R-spondin1 (both 10%; conditioned medium prepared in house), Y-27632 (1 µM) and 4 µg ml−1 DNase I (Sigma-Aldrich). The cell suspension was filtered through a 70 µm cell strainer (Celltrix) and analysed using flow cytometry on the FACS Aria Fusion (BD Biosciences) system. Dead cells were eliminated by gating of forward/side scatter, forward/pulse-width parameters and negative staining for 7-AAD (eBioScience). LGR5+ cells were sorted by positive staining for Epcam (CD326) (BD Bioscience, 563214) and endogenous LGR5–GFP signal. Data were visualized using FlowJo v.10.6.1 (https://www.flowjo.com/).

For RNA-seq, sorted cells were collected into 1.5 ml tubes containing FACS buffer, washed with PBS, pelleted and stored at −80 °C for further processing. RNA was isolated using RNeasy Micro Kit (Qiagen).

For colony-formation assays, sorted cells were collected in FACS buffer, washed with PBS, pelleted and embedded in Cultrex PathClear Reduced Growth Factor Basement Membrane Extract Type 2 (BME2) (Amsbio, 3533-005-02), followed by seeding onto 24-well plate (500 cells in 20 μl of BME2 per well). The number of organoids were counted 7 days after seeding.

Processing RNA-seq expression data

For library preparation, the SMART-Seq Stranded Kit (Takara) was used. Sequencing was performed by CeGaT on the NovaSeq 6000 system with 100 bp paired-end reads. Demultiplexing of sequencing reads for all 18 samples (three biological replicates of LGR5+ high, medium and low cells for both SI and LI) was performed using Illumina bcl2fastq (v.2.20) for which adapters were trimmed using Skewer (v.0.2.2)26. Trimmed raw reads (average length 96–102 nucleotides) were aligned to the mm10 genome using STAR (v.2.5.2b)27 for genome assembly and gene count.

Differential expression analysis was performed between groups considering biological replicates of intestinal locations using DeSeq2 (v.1.34 in Bioconductor v.3.14)28 in R (v.4.1.1; R Core Team 2021). After normalization, all regions with a mean count greater than zero were included to improve detection power. The mean count across genes above zero was selected based on the standard deviation for each genes. With normalized counts we calculated the log2-transformed fold change and obtained the P value (Wald test) and P value adjusted using Benjamin–Hochberg correction for multiple testing.

The relationships between intestinal locations on the basis of gene expression were visualized through principal component analysis. We used regularized logarithmic transformation (rlog) of the normalized counts for all samples to obtain the principal component analysis so each gene contributes equally to the distance between samples. Volcano plots were obtained in R only highlighting differentially expressed genes with P < 0.001 and a log2-transformed fold change higher or lower than 2 and −2 for upregulated and downregulated genes, respectively.

Preparation of decellularized mouse intestine

Small intestinal tissue was decellularized as previously described15. Pieces of decellularized small intestinal ECM (dECM) were washed for 10 min in PBS, plated on a culture dish, and stained with Col-F Collagen Binding Reagent dye (Col-F) (BioSite, 260-6346) overnight at 4 °C. dECM was then washed with fresh PBS, and primed for approximately 30 min with live imaging medium consisting of Advanced DMEM/F12, 1× penicillin–streptomycin, 1× Glutamax (Thermo Fisher Scientific), 10 mM HEPES (Thermo Fisher Scientific), 1× B27 (Life Technologies), 1× N2 (Life Technologies), 1 mM N-acetylcysteine (Sigma-Aldrich), 50 ng ml−1 of murine recombinant epidermal growth factor (R&D), 100 ng ml−1 recombinant murine Noggin (Peprotech), 10 µM Y-27632 (Sigma-Aldrich) and 1 µM jagged-1 peptide (Anaspec). After priming, excess medium was discarded to enable seeding of LGR5+ cells.

Single-cell isolation and FACS

LGR5+ cells were isolated as previously described20, with a modified protocol to effectively detach epithelium. In brief, longitudinally opened intestine was cut into 5–7 cm pieces, and incubated in 10 mM PBS-EDTA for 30 min. Intestinal pieces were first gently scraped using a microscopy slide to discard villous material and scraped once more to collect crypts.

Live imaging of LGR5+ cells on dECM

LGR5+ cells were stained and washed with 5 μM 647CellTracker (Invitrogen) according to the manufacturer’s protocol. Approximately 10,000 cells in 10–20 µl of medium were then seeded on to the scaffold and were allowed to settle for 10 min before adding 10 µl of medium. Additional medium was then carefully introduced in two phases: 20 µl at 20 min after seeding and 260 µl at 30 min after seeding.

Cells were live imaged with a Nikon spinning-disk confocal microscope, using a PlanApo ×10/0.5 NA dry objective and NIS-Elements AR (v.5.02.01; Nikon). A stack of images covering the whole height of the crypt–villus axis was captured with 2.5 µm separation between the images, and the full stack was imaged every 2 min. Cells were imaged for about 4 h and live/dead cells were assessed at the end of imaging by staining with DAPI (Thermo Fisher Scientific, 1 μg ml−1). Col-F was detected by using a 488 nm laser with a 520/30 emission filter, the 647CellTracker was detected using a 640 nm laser with a 685/40 emission filter, and a 405 nm laser with an emission filter 447/60 for DAPI.

Analysis of single-LGR5+-cell migration on dECM

For preprocessing, the images were corrected for xy drift, when needed, using the ‘Register Virtual Stack Slices’ and ‘Transform Virtual Stack Slices’ plugins in ImageJ. In brief, the bright-field images were used to make a maximum intensity projection over the z-levels. Next, drift was registered using with the following parameters: ‘Feature extraction model: Translation’; ‘Registration model: Translation --no deformation’; ‘Save transforms’ option selected. Subsequently, the transformations over the time lapse were applied for each of the respective z-level on the bright-field and fluorescent images using the ‘Transform Virtual Stack Slices’ plugin, which were then reassembled into hyperstacks.

Cell tracking was performed with a combination of automatic tracking, using the ‘Trackmate’ plugin (v.6.0.3)29 in ImageJ, and manual adjustment of the selected tracks (for example, joining split tracks as needed). The results were then exported for subsequent analysis.

Analysis of the cell tracking results was performed using custom scripts in Python (v.3.10). In brief, each cell was classified on the basis of its location in z (acquired by visual inspection of the decellularized intestinal pieces before the start of the time lapse) into crypt or villus regions. Cell motility was calculated as the Euclidean distance between each time point and its preceding time point. Finally, the results were plotted as the mean and 95% CI, using a polynomial fit (numpy.polyfit, numpy v.1.19.5) for each of the tracks.

In vitro migration assay

Small intestinal organoids from the Lgr5-EGFP-ires-creERT2 × R26R-confetti mouse were generated and directly induced (that is, before the first passage) with TAT-Cre recombinase (5 μl ml−1, Merck, SCR508). After 5 days in culture both, the LGR5+confetti–RFP+ and LGR5+/confetti–YFP+ stem cells were sorted (SONY MA900 cell sorter) and plated as single cells30. After 7 days of culture, 2,500 or 1,250 LGR5–GFP+confetti–RFP+ stem cells and 1,250 CD24high/confetti–YFP+ PCs were sorted from these two respective organoid cultures14 into ultra-low-attachment 96-well round-bottom plates (Sarstedt, 82.1582.001) containing 135 μl of medium. PCs were sorted on the basis of CD24 expression (1:400, Thermo Fisher Scientific, 48-0242-82) and all cells were exposed to live–dead dye (1:200, DRAQ7, Thermo Fischer Scientific, D15105) to sort for living cells. The basic medium25 was supplemented with either 50% WNT3A CM, IWP2 (2 µM, Tocris, 3533)31. After sorting, the plate was left on ice for 15 min and 15 μl of Matrigel was added before 5 min of centrifugation (300g with low acceleration and breaking). The fluorescent and bright-field images were acquired every 15 min by a spinning-disk confocal (ImageXpress Micro Confocal) equipped with a ×10/0.45 NA Nikon Plan Apo Lambda objective and live imaging chamber (humidified with sterile water and maintained at 37 °C, 6.4% CO2). All organoid lines were tested and confirmed to be negative for mycoplasma contamination.

Processing of in vitro migration data

The TrackMate plugin (ImageJ v.2.3.0)29,32 was used to unbiasedly track the location of individual cells over time (estimated object diameter = 10 μm, linking max distance = 45 μm, gap closing max distance = 15 μm, gap closing max frame gap = 1). CelltrackR33 (R v.4.1.1), DiPer34 (Microsoft Excel 2016) and custom made code were used to calculate the mean square displacement, speed and persistence. Replicates with a low number of tracks (<40) or for which the cell density was near confluent were excluded from the analysis, leading to the exclusion of one biological replicate. For each condition, the data are based on three independent biological replicates, each including two to four technical replicates. To calculate the migration speed and directionality of the motile cells, only tracks from motile cells were included (that is, with speed >0.3 μm min−1).

DT treatment

For the LGR5+ cell ablation studies, male and female Lgr5DTR:eGFP mice received 50 μg kg−1 DT in PBS through intraperitoneal injections for 2 days in a row and were euthanized after 24 h, 48 h, 96 h, 7 days and 15 days. Small and large intestines were collected, stained with DAPI and Phalloidin Alexa Fluor 647 (Thermo Fisher Scientific, A22287) and imaged ex vivo.

Statistics

Any a priori sample size calculation could not be performed, as the effect size and the variance was not known before the experiments. No data or animals were excluded, except for in the in vitro migration assay, in which one biological replicate was excluded owing to low number of trackable cells (<40). Owing to differences in crypt morphology, blinding was not possible when comparing the SI and LI. All other comparisons of intravital microscopy experiments and the in vitro migration assay were analysed in a blinded manner. The imaging data were randomized by one researcher and analysed by another researcher. For the analysis of the migration data, the randomization was performed using a custom script in Python and R, and analysed in a blinded manner by the researcher. All P values were calculated using two-sided Mann–Whitney U-tests in GraphPad Prism v.9 (GraphPad). Details on statistics concerning the mathematical modelling are provided in the Supplementary Notes.

Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this paper.