Measuring morphogen transport over multiple spatial scales in live zebrafish embryos

Abstract

Morphogenesis, the development of multicellular organisms, is controlled by morphogens, small signalling molecules, that form gradients across the embryo. While such gradients can emerge through several mechanisms, the establishment of stable gradients requires the reduction of morphogen diffusion by at least one order of magnitude, a process that has not been directly observed. Here, we develop Single-Plane Illumination Microscopy-based spatial Fluorescence Cross-Correlation Spectroscopy (SPIM- sFCCS) to directly measure the diffusion coefficient of the morphogen Squint in early zebrafish embryos as a function of topography and length scale. The diffusion coefficient of Squint is a function of its molecular size, the viscosity within the interstitial spaces, and possible crowding. Here, we show that beyond these general parameters, Squint’s diffusion coefficient changes on length scales that are commensurate with the diameter of the intercellular spaces in the embryo and that its slowdown is regulated by receptor binding, which is crucial for gradient formation and embryonic patterning.

Morphogenetic gradients can emerge through the slowdown of morphogen diffusion, however, the precise monitoring of this process across micrometre scales remains difficult. Here, the authors present Single-Plane Illumination Microscopy-based spatial Fluorescence Cross-Correlation Spectroscopy (SPIM-sFCCS) to measure the diffusion of the morphogen Squint in early zebrafish embryos as a function of topography and length scale.

Introduction

At the heart of morphogenesis are concentration gradients of morphogens that control cell fates and patterns^1,2. The movement of morphogens is regulated by their interactions with cell surface receptors³ and the extracellular matrix^4,5. These interactions trigger the expression of key downstream genes, activators, inhibitors, and degradation pathways, all of which shape the morphogen gradient over space. The mobility of these signalling molecules is thus a limiting factor that dictates the rate at which signals propagate over space in an embryo, and when and where biochemical signalling pathways and developmental processes are carried out. Therefore, maintaining robust transport dynamics of morphogens across space is vital for normal developmental events to take place at the right place and time.

To convey specific positional information to different cellular locations, morphogens are transported across space either through passive diffusion⁶ or active transport (transcytosis^7–9 or cytonemes¹⁰).

In the case of diffusion-based transport, the span of the gradient is dictated by the interplay between diffusion and the degradation rate of the ligand. The gradient length is $\sqrt{D/k}$, where D is the diffusion coefficient, and k is the degradation constant of the ligand. While interactions with ligand inhibitors or other morphogens and inherently slow diffusion of bulky, tumbling molecules also influence the gradient formation, these mechanisms act to modify the effective transport or removal rates.

Morphogens are signalling ligands of low molecular weight and, accordingly, have high D values (10–100 μm²/s). Moreover, measured degradation rates in vivo, as far as available, are low (~10⁻⁴/s)^11–13. These parameters would result in an ineffective gradient length much larger than the embryo. Accordingly, global morphogen diffusion coefficients (D_glob) measured over a large region (> 100 μm) are lower by one to two orders of magnitude compared to their local diffusion coefficient (D_loc). This slowing down of morphogens from the local to global scale is seen in several morphogens, including Nodals, Wnts, BMP, and FGF¹⁴ (and references therein) and is crucial for the establishment of robust concentration gradients over space¹⁵.

Several factors that cause ligand mobility to slow down have been demonstrated in vivo. Tortuosity, i.e., the hindrance of movement of particles due to the presence of cells, has been shown to slow down molecules in tissues¹⁶, ⁵ (and references therein). Reduced mobility by either binding to the extracellular matrix elements or being physically hindered by their presence has also been shown to contribute to the slowdown of morphogens^4,17. Moreover, transient receptor binding has been demonstrated to hinder morphogen movement by slowing down its effective mobility across space^13,17. In fact, it was demonstrated that tuning receptor concentrations in synthetic ligand-receptor pairs can influence the global diffusivity of ligands¹⁸ and that receptor binding-based hindrance was sufficient to form synthetic concentration gradients of ligands in vivo (refs. ^19,20). Additionally, the geometry of extracellular space regulating the binding propensity of ligands to receptors was also demonstrated using Single Particle Tracking (SPT) in vivo²¹.

Although the reasons for the slowdown are well established, the extent to which each factor contributes to the slowdown and the exact mechanism of how these factors combine to hinder diffusion need to be adequately addressed. Moreover, the length scales at which each of these factors operates to produce the transition from fast to slow diffusion need to be established. Probing the rate and scale of the transition from molecular size-dependent mobility to spatial interaction-hindered mobility allows a greater understanding of the rate of signalling processes that occur at characteristic length scales and helps model developmental processes.

To understand the extent and the length scale at which the slowdown occurs, we need to map the transition of morphogen diffusion from the local to the global scale. However, the commonly used in vivo morphogen diffusion measurement techniques do not cover all spatial scales. The D_glob measured using Fluorescence Recovery After Photobleaching (FRAP)^11,22,23 quantifies the recovery rate of fluorescence caused by diffusing fluorescent molecules into a large photobleached region of interest (10–100 μm). It is thus affected by the tissue morphology or tortuosity, and also by interactions of the morphogens with components of the extracellular space and cells (Fig. 1a–c). D_loc, on the other hand, is measured by single-molecule sensitive techniques. For example, Fluorescence Correlation Spectroscopy (FCS) in vivo^4,24,25, quantifies the time-span of the fluctuation of fluorescence signals arising from molecules diffusing in and out of a confocal volume, which is limited to < 1 μm. It thus recovers the unencumbered free diffusion coefficient determined only by particle size, viscosity, temperature, and possibly interactions with the extracellular matrix (Fig. 1c). SPT follows single molecules along their trajectories in time with excellent spatial resolution. It thus could, in principle, follow single molecules on their trajectory. However, due to the limited photostability of fluorophores, the single-molecule trajectories rarely exceed several microns. Neither of these techniques can fully bridge the gap between the local and global scale, and thus the intermediate length scales (1–10 μm) are hard to probe. Therefore, to determine the scale of influence of these diffusion regulators and probe the scale-dependent transition, there is a need to bridge the spatial information gap at the mesoscale (< 10 μm) between the local FCS and the global FRAP measurements (Fig. 1b, d).

Fig. 1. Hindered diffusion across different length scales.

a An illustration of the tissue space at the scale of 100 μm. FRAP measurements (area covered by the circle) are performed at this length scale, and they capture the combined influences of tortuosity and receptor binding to hinder the diffusion of morphogens. b An illustration of the zoomed-in space of the tissue (square in this figure (a)) capturing a single cell and its surrounding environment at the length scale of 20–30 μm. Morphogens diffuse and interact with the cell surface receptors in the interstitial spaces between cell membranes. c An illustration of the zoomed-in space of the interstitial space (circle in this figure (b)) capturing information from a diffraction-limited point in space (< 1 μm), as dictated by the observation volume of a confocal microscope. Morphogens may diffuse freely or interact with the ECM at this length scale, which can hinder their mobility at the local scale. d An illustration of the interstitial space zoomed in (rectangle in this figure (b)), capturing the inter-membrane space between two cells at a scale of 10 μm. Morphogens can bind to the receptors on the membrane surface and transiently halt, hindering their diffusion across the mesoscale. e An illustration of the application of the spatial cross-correlation approach to capture information about morphogen transits across two specific points within the inter-membrane space from this figure (d). The red and blue cross-correlation function (CCF) curves correspond to the respective transits of molecules measured from white to red and white to blue boxes, respectively. The peak in the CCF captures the mean transit time of morphogens diffusing between a pair of points in space. The peak shifts to longer lag times with increasing separation distances since molecules take longer times to diffuse farther in space. Additionally, there is a reduction in correlation amplitude due to the reduced probability of finding the same set of molecules starting out from the initial point being considered for the spatial cross-correlation. Created in BioRender. Nelanuthala, A. (2026) https://BioRender.com/u57r906.

In this work, we address this information gap by probing the mobility transition from local to global length scales using spatial cross-correlations in Imaging FCS. Imaging FCS is an FCS modality which illuminates a thin cross-section of a sample and, through fast camera readout, records intensity fluctuations caused by molecular movement for each pixel. By calculating and evaluating the pixel autocorrelation functions, it provides spatially resolved maps of diffusion coefficients²⁶. Imaging spatial Fluorescence Cross-Correlation Spectroscopy (Imaging sFCCS) can be implemented with the same method, but instead of auto-correlating the signal of each pixel, one cross-correlates pixels at different distances. Imaging sFCCS thus captures the transit time of molecules diffusing or flowing between any two points within the image (Fig. 1d, e). sFCCS has previously been established either by two-focus FCS^27–29 or scanning FCS^30–33 to measure transit times between two distinct points in space and thus obtain the diffusion coefficient or flow velocity of the molecule of interest. However, the information on spatial dynamics obtained from such techniques is limited to either two specific points or across a certain line in sample space. sFCCS performed on imaging-based modalities like pair correlation function analysis^34–37 and Imaging FCS^38,39 allows for a wider spatial range for capturing dynamics between any set of pixels. It thus helps obtain information about the cumulative influence of molecular interactions and barriers to diffusion in different directions of space, allowing one to correlate molecular mobility with the underlying microstructure of the environment. Coupled with camera-based FCS⁴⁰, it has been used to probe space-dependent heterogeneity in diffusion dynamics in vivo^41–43.

The application of sFCCS on light sheet imaging data has already been used in the past to probe direction-dependent diffusion and hindrances on 2D cell membranes⁴⁴. Here, we demonstrate the application of sFCCS in 3D within a single-plane illumination microscope (SPIM-sFCCS) to probe effective diffusion coefficients (D_eff) over space in 3D environments. We modify the current 3D FCS fitting models of SPIM-sFCCS to account for signal crosstalk between pixels in a light sheet microscope for accurate D_eff estimation over space.

We apply SPIM-sFCCS to study the dynamics of the zebrafish Nodal morphogen Squint (Sqt) to capture its scale-dependent D_eff. Nodal signalling is essential for mesendoderm induction and left-right patterning in early vertebrate embryos^45,46. The dynamics and concentration gradients of Nodals have been extensively characterised due to their important role during gastrulation. Sqt is a morphogen since it can induce the expression of different genes across space in a concentration-dependent manner⁴⁷. Sqt exhibits a D_loc of 40–60 μm²/s^5,48 as measured by confocal FCS, and D_glob drops to as low as 2 μm²/s when measured using FRAP.^11,16.

While Sqt may not bind to the extracellular proteoglycans due to its acidic residues giving it added stability and a long decay rate^11,48,49, it does bind to cell surface receptors Acvr2b⁴⁸, Acvr 1a/1b¹³, and the co-receptor Oep⁵⁰. It was shown that overexpression of Oep increased the bound fraction of Sqt²¹ and further reduced D_glob¹³ and knockdown of a combination of receptors (Acvr 1b-a/b), increased Sqt’s D_glob¹³.

Nodal signalling is active across 3–6 h post fertilisation (hpf)^51,52 in a time window where the blastula tissue has large cells and when several tissue-level transitions occur that create different tissue densities⁵³. Unlike in denser tissues seen at later stages, the blastula tissue, with visibly wide and varying intercell spaces available for measurement, makes it an amenable model environment for studying the mechanism of morphogen slowdown using SPIM-sFCCS.

Here, we first use diffusion simulations in channel-like spaces to understand and establish how the extent of cell–cell spacing and receptor binding can influence the scale and the value of D_eff. Finally, we validate the simulation results experimentally. We measure D_eff of Sqt diffusing in the interstitial spaces between cells across different lengths and show that Sqt slows down and reaches a steady state diffusion coefficient value only at a distance greater than the intercell spacing. Moreover, knocking down the receptor Acvr2b (The paralog used here is Acvr2b-a and was identified previously¹³) or overexpressing the inhibitor Lefty2, increases the steady state D_eff of Sqt to a value similar to secreted EGFP, an inert non-interacting protein, implying that receptor binding hinders Sqt diffusion.

Results

Receptor binding simulations in silico

To effectively employ SPIM-sFCCS to probe the cause of morphogen slowdown across space, we first identified the physicochemical factors that influence the deceleration of morphogen diffusion. To that end, we simulated the mechanism of morphogen slowdown incorporating reversible receptor binding. We modelled the spaces between cells as channels of a certain width with the channel running along the x-axis. While the interstitial spaces in vivo may range from < 100 nm in tight junctions to ~10 μm (comparable to cell diameter) in wide intercell spaces, we used channel widths ranging from 0.5 to 5 μm to model and understand the diffusion with binding behaviour in spaces similar to where we measure.

Simulations were run with a time step of 0.01 ms for 100,000 steps, resulting in 1 s of simulated time. All simulated particles started across the middle of the channel at position (x, y) = (0,0). The particles spread with time, and when encountering the channel wall, could bind or be reflected depending on a defined binding probability (set to 0.9 if not otherwise mentioned). Bound particles were allowed to unbind with a defined unbinding probability (set to 0.01 if not otherwise mentioned). These two probabilities define the receptor affinity. D_eff was calculated in regular intervals from the root mean squared displacement (RMSD) of all particles in the x-direction (Illustration in Fig. 2a).

Fig. 2. Simulations of binding-hindered diffusion in the inter-membrane spaces between cells.

a Illustration of how particles diffuse and bind to the walls of a channel in simulations. The effective diffusion coefficient over a certain period is calculated based on the distribution of particles across the length of the channel. (Top illustration created in BioRender. Nelanuthala, A. (2026) https://BioRender.com/e2u602n, bottom illustration created in BioRender. Nelanuthala, A. (2026) https://BioRender.com/r2jcubd). b Plot of the transition of effective diffusion coefficients over distance for different channel widths. The effective diffusion coefficient drops from the local value and reaches the steady-state value only after a certain distance dictated by the channel width. c Plot of the effective diffusion coefficients obtained from different simulated channel widths and binding probabilities at a fixed distance of 3.6 μm (to compare with the results of Fig. 5b, c). Error bars are the SD of values from 3 independent simulation seeds. The solid lines are guides to the eye through the mean values. If the ligands experience a binding-based hindrance, the effective diffusion coefficient obtained at a steady state is channel-width dependent. Source data for this figure (b, c) are provided in the Source Data file.

For an illustrative case with constant binding probability (0.9) and unbinding probability (0.01), channels of different widths had different effective diffusion coefficients at steady state, with smaller channels leading to a decrease in D (5 μm: 1.5 fold, 0.5 μm: eightfold; see Fig. 2b). In all cases, D_eff converged to a steady state relatively quickly (within 0.01–0.2 s of the simulation time), with particles in narrow channels reaching a steady state quicker than in wider channels (Supplementary Fig. 1a). In the spatial domain, D_eff converged to a steady state value only at a distance greater than the width of the channel (Fig. 2b). Overall, simulations showed a channel width-dependent diffusion coefficient (Fig. 2b, c).

Progressively varying the binding probabilities from 0 to 0.9 while keeping the unbinding probability constant (0.01), reduced D_eff at steady-state (Supplementary Fig. 1b left and c left). For the 1 μm channel, the highest binding probability set at 0.9 caused a slowdown of almost fivefold. For the case of 0 binding probability, that is, when there was no binding, D_eff remained unchanged. Conversely, reducing unbinding probabilities from 0.1 to 0.001, reflecting increased ligand-receptor affinity, significantly decreased D (Supplementary Fig. 1b right and c right), with the extent of slowdown reaching almost 50-fold for the combination of 0.001 unbinding probability, 0.9 binding probability, and 1 μm channel width. Even a binding probability of 0.1 coupled with an unbinding probability of 0.001 resulted in almost a threefold reduction in D (Supplementary Fig. 1c right). Different combinations of binding and unbinding probabilities yielded similar D_eff values for a given channel width, demonstrating the coupled effect of binding and unbinding rates influencing diffusion. Zero unbinding probability did not allow for particle dispersal over space, emphasising the necessity of reversible binding for ligand transport.

Overall, these simulations demonstrated that the geometric confinement of channel-like spaces facilitates reversible binding to slow down the D_eff of particles across space. Based on these simulations, measuring D_eff across a certain distance in a channel should yield a channel width-dependent D_eff value in the presence of receptor binding. Reducing the binding levels should increase D_eff, and in the case of no binding, D_eff should remain unchanged irrespective of channel widths (Fig. 2c).

SPIM-sFCCS

First, before applying SPIM-sFCCS in tissues, we derived a new theoretical fit model for data analysis. This became necessary as the crosstalk between the contiguous camera pixels leads to pseudo-autocorrelations that need to be taken into account for correct data evaluation (Fig. 3a, b).

Fig. 3. Addressing the issue of signal crosstalk across space in SPIM-sFCCS.

a Illustration of SPIM's observation volume and how the overlap of the diverging sections of the observation volume in the axial direction causes an overlap in the detection volumes of a pair of pixels across space on the camera’s detector, resulting in signal crosstalk. Created in BioRender. Nelanuthala, A. (2026) https://BioRender.com/lrs2trk. b Comparison of the modified fitting model (solid lines) and the previous model (dashed lines) for average cross-correlation functions (points) for different separation distances obtained from simulated data. The separation distance is given in pixels (1 pixel = 0.4 μm). The modified fitting model is able to capture the crosstalk of signal between pixels at short separation distances, unlike the previous model. c Comparison of the average cross-correlation function (points) from multiple correlations across space and the corresponding fits (solid lines) using the modified fitting model for different simulated diffusion coefficients for a certain separation distance. The modified model is able to capture the extent of the signal crosstalk component in the cross-correlation function for a range of diffusion coefficients tested. d Fit data of the diffusion coefficient obtained from the average cross-correlations over space in simulated 3D free diffusion data with different diffusion coefficients. Error bars: SD of three independent simulations. The solid lines are guides to the eye through the mean values. Dashed lines are ground truth values. The modified fitting model can capture accurate diffusion coefficients across a certain distance for a range of different diffusion coefficients. e Comparison of the auto and cross-correlation functions (blue) obtained from correlating multiple points in space and their corresponding fits (red) for different separation distances from freely diffusing fluorescent beads in water (N_acf  = 84, N_ccf  = 63, 56, 49 for x = 3, 4, 5 respectively; See Supplementary Movie 1 for the timelapse). The modified fitting model implemented on the GPU can capture the crosstalk of signal between pixels at short separation distances and provide constant diffusion coefficient values across space. Source data for Fig. 3b–e are provided in the Source Data file.

It is known that in multi-foci confocal FCS, overlapping observation volumes result in signal crosstalk between detectors⁵⁴. The crosstalk, in this case stems from the laser beam which propagates through the whole sample in the axial direction and produces out-of-focus fluorescence that is detected by neighbouring observation volumes^27,55. When cross-correlating neighbouring pixels, a pseudo-autocorrelation is seen as the pixels detect the same molecules at the same time. By enforcing a minimum distance between observation volumes, this crosstalk can be minimised.

In SPIM-sFCCS this problem is less severe as the light sheet has a finite thickness and thus less out-of-focus fluorescence is produced. Thus, the minimum distance between pixels that show no pseudo-autocorrelations is smaller. Nevertheless, as we use contiguous pixels in SPIM-sFCCS, the crosstalk between pixels can span over considerable distances and lead to a pseudo-autocorrelation component in the correlation functions (Fig. 3a). In our SPIM system, this signal crosstalk can be detected over a pair of pixels spaced almost 6 pixels (2.4 μm) apart (Supplementary Fig. 2a).

The extent of crosstalk increases with an increase in the span of PSF_xy due to a larger overlap in observation volumes (Supplementary Fig. 2b). The extent of crosstalk also increases with the thickness of the light sheet due to the increase in out-of-focus contribution from thicker light sheets (Supplementary Fig. 2c). However, the effects of signal crosstalk are not significant at larger separation distances (Supplementary Fig. 2d).

The fit model developed by ref. ⁴⁰ did not account for the excess crosstalk, and the change in the PSF_xy along the z-direction. While it is adequate for fitting autocorrelations, it cannot fit the CCFs.

We, therefore, developed a modified SPIM-FCS fit function based on ref. ²⁹ that accounts for this crosstalk by incorporating PSF_xy as a function of z, similar to the model for the confocal’s observation volume in ref. ⁵⁶. The fit function’s derivation for arbitrary separation distances and pixel bins in x and y (Supplementary Fig. 3) can be found in Appendix C. Since no analytical solutions could be found, the fit model was numerically integrated along the z-axis. The step size used in the numerical model was optimised to accurately fit ACFs and CCFs with D > 0.5 μm²/s (Appendix C–Supplementary Fig. 4). As fitting with numerical models is computationally expensive, we implemented data fitting on a GPU, improving evaluation time 60-fold over CPU-based fitting.

The fit model was tested on 3D simulations for different distances and different D values. The model was able to capture the additional crosstalk component for different distances (Fig. 3b) and different diffusion coefficients (Fig. 3c). The fit model also gave accurate results within 20% of the simulated D value for auto/cross-correlation data (Fig. 3d), with the spatial range for accurate fits depending on the D value being measured (discussed in optimising SPIM-sFCCS for in vivo measurements section).

Applying the model to fit spatial cross-correlations from SPIM-sFCCS measurements of fluorescent beads in solution (Supplementary Movie 1) yielded constant D across space, within 20% of the theoretically expected value while accounting for the increased pseudo-autocorrelation component due to crosstalk (Fig. 3e). These results confirmed the model’s ability to accurately capture D_eff across 3D space.

Optimising SPIM-sFCCS for in vivo measurements

Before applying SPIM-sFCCS in vivo, we developed data evaluation strategies to optimise the signal-to-noise ratio (SNR). We achieve this through spatial and temporal binning of pixels post-acquisition, where the binning range is determined by the particular range of concentrations and diffusion coefficients to be measured, and by the limited measurement time available due to tissue dynamics of the embryo. In FCS, the SNR increases with the number of molecular transits recorded^57,58. Therefore, increasing measurement time reduces the error of fit parameters and allows for accurate CCF fits over longer distances (Supplementary Fig. 5a, b). Similarly, increasing diffusion coefficients allows for more particle transits to be captured, which also improves the signal for correlations (Supplementary Fig. 5c) and increases the fitting range (Fig. 3d and Supplementary Fig. 5d). This observation is advantageous for the current work since zebrafish ligand diffusion at the local scale ranges from 10 to 100 μm²/s¹⁴. However, for high Ds (60 μm²/s in Fig. 3d), estimations over short distances (~0–3 pixels) have poor accuracy due to the limited temporal resolution of the camera (~1 ms).

Owing to the dynamic nature of the zebrafish blastula between 3 and 5 hpf, measurements with stationary interstitial spaces and constant inter-membrane spacing are limited in duration (30–120 s at 4 hpf, < 20 s at 5 hpf). These measurement times are insufficient to provide adequate transit statistics for SPIM-sFCCS measurements.

To improve the signal statistics for SPIM-sFCCS post-acquisition, we binned pixels around the two spatial regions of interest. Pixel binning increases the number of molecular transits being captured between the two points (Fig. 4a and Supplementary Fig. 5e). This, in turn, improves the quality of the cross-correlation functions (Fig. 4b, c) and the accuracy and precision of the measured D (Fig. 4c and Supplementary Fig. 5f).

Fig. 4. Signal improvement strategies for implementing SPIM-sFCCS in vivo.

a Imaging-FCS allows binning signals from multiple pixels in different ways to perform SPIM-sFCCS. b Comparison between the average CCFs calculated with rectangular and square pixel binning from simulated data. Square binning causes the detection volume of pixels to overlap, causing higher crosstalk than rectangular bins. c Comparison of the raw CCFs (blue) and corresponding fits (red) calculated with different pixel bin sizes from simulated data (N_ccf  = 280 (left), 154 (right)). Increasing binning improves fits and reduces SDs of the diffusion coefficient. d Consecutive frames in the time-series image stack can be binned as an alternate way to increase exposure post-acquisition. e Comparison of the average CCF amplitude across different simulated signal levels. The 1 and 5 ms exposure data were from simulated stacks with 100,000 frames and 20,000 frames, respectively and N_noise = 3. The amplitude increases with exposure and reduced noise. The dashed curve is the average CCF of a 1 ms exposure time stack, binned 5× in time (5 ms time resolution). This increases CCF amplitude, but not as significantly as with increased exposure time. f Example of frame time-binning obtained from an in-vivo Sqt-EGFP measurement (raw correlations: blue, fits: red). Left: Spatial CCFs from raw data. Right: CCFs of the same image stack after 4× frame-binning (N_ccf  = 8 for both). Binning increases the amplitude of the CCFs without affecting CCF peak positions. g Comparing CCFs in the forward and reverse directions. Equal forward and backward values ensure that the dynamics captured are diffusion-like and free from drift, flow, or other active dynamic components. h Example of an ROI (region of interest) from an embryo expressing Sqt-EGFP (see Supplementary Movie 2 for time-lapse) to illustrate forward-backwards equivalence of measured effective diffusion coefficients. i Spatial CCF data obtained in the forward and reverse directions (raw correlations: blue, fits: red, N_ccf  = 20 for both) from the data shown in this figure (h). Similar values indicate that the measured diffusion coefficients capture the true diffusive component of the dynamics without cell movement-based artefacts. Source data for this figure (b, c, e, f, i) are provided in the Source Data file.

The amplitude of the correlation function decreases with increasing noise levels (Supplementary Fig. 5g) and increases with higher counts per molecule (Supplementary Fig. 5h). To improve SNR post-acquisition, we increased the acquisition exposure by binning frames in time (Fig. 4d). The correlation amplitude increases, although the improvement is not to the same extent as when acquiring with an equivalent increased exposure per frame (Fig. 4e). Nevertheless, this approach not only allows the use of short and long exposures from a single measurement but also reduces the bias from fitting noise at small lag times for long-distance measurements on experimental data (Fig. 4f).

To capture high-speed dynamics for in vivo measurements, (see “Methods”/Supplementary Fig. 6 for measurement workflow) we acquired signals with the fastest frame rate available (0.5–2 ms exposure corresponding to 2000–500 frames/s) for the measurement depending on the size of the ROI selected and with higher laser power (60–600 W/cm²) compared to imaging (30–60 W/cm²) to capture sufficient counts per molecule. For long-distance measurements (3.6–5.2 μm), where cross-correlation amplitudes were low, we binned frames in time to obtain image stacks of 2–6 ms exposure time to increase SNR for improved data fitting.

Cellular movements can create convective flows that lead to deviations from simple diffusion (Supplementary Fig. 7a). As convective flows are directional, the CCFs will depend on the direction for which they are calculated. Only if the forward and backward CCFs provide the same diffusion coefficient can one assume the absence of flow or directed transport (Fig. 4g). Discrepancies between forward and backward D values indicated deviations from simple diffusion (Supplementary Fig. 7b–d) and such data were excluded.

Similarly, in the presence of barriers around the interstitial spaces, molecules diffusing into the barriers get reflected (Supplementary Fig. 7a). These reflected molecules contribute twice to the transit statistics, resulting in a wider CCF measured in the direction of the barrier compared to the one away from the barrier (Supplementary Fig. 7e–g). The approach of checking the equivalence of forward and backward D values helps eliminate such effects.

In cases where the channel drifts along the z-axis, the forward and backward D may appear equal. However, the diffusion coefficient values are underestimated since the molecules take longer to transit across two points. Moreover, while we ensured that channel spacing remained constant within the analysis time window, we recognise that subtle changes in channel widths or the influence of the invisible axial spaces can lead to inaccurate estimations even if the bidirectional D values match.

We compared the mean forward and backward D_fit values and ensured that the difference was within 30% of the mean D value (for an example of this approach, see Fig. 4h, i). Across all the included datasets, the mean difference between D_fwd and D_bwd was 11.4% of the mean D value.

We found that at least 30% of all measured data had significant tissue movements, resulting in CCFs that could not be evaluated. 40% of measurements showed differences in the forward and backward diffusion coefficient, indicating convectional flow due to tissue movement. The remaining 30% of the measurements, which showed equivalent forward and backward Ds, implying purely diffusive transport, were only considered.

Distance-dependent morphogen diffusion in zebrafish embryos

The results from the simulations of particle diffusion in channel-like spaces with receptor binding suggest that the slowdown and the resultant steady-state D_eff are channel-width-dependent, with the transition happening at a length comparable to the inter-membrane spacing between two cells. Having modified the fitting model and accounted for low signal and cell movements, we sought to verify the simulation results’ applicability to morphogen dynamics in vivo.

We initially compared D_eff of Sqt over a fixed distance of 9 pixels, corresponding to 3.6 μm, for channel-like spaces with different inter-membrane spacings. In Fig. 5a, b, it can be seen that for narrow spacings, D_eff is low, with the lowest spacing giving ~10 μm²/s. With increasing channel width, D_eff increased up to a spacing of 4 μm, after which it plateaued at ~51 μm²/s. Furthermore, to compare the effect of the channel width and the extent of slowdown experienced in narrow spaces, we grouped the results from narrow spaces and wide spaces (Fig. 5c). We found that there was almost a threefold drop in D_eff for narrow channels (< 2.5 μm: ~18 μm²/s) compared to channels with the width of inter-cell spacings > 4 μm (~51 μm²/s) (p < 0.0001).

Fig. 5. sFCCS measurements in channel like spaces in vivo.

a Comparison between D_eff measured over 3.6 μm in wide and narrow channel-like spaces in Sqt-EGFP expressing embryos (raw correlations: blue, fits: red, N_ccf  = 80 for wide, 13 for narrow) (See Supplementary Movies 3 and 4 for time-lapse videos). b A comparison of D_eff measured at a separation distance of ~3.6 μm between sec-EGFP: (N_ch = 16, N_emb = 8), Sqt-EGFP: (N_ch = 24, N_emb = 18), Sqt-EGFP + Acvr2b Morpholino: N_ch = 15, N_emb = 9, Sqt-EGFP + Lefty2: N_ch = 16, N_emb = 8) from channels of different sizes (widths). The error range (shaded areas) represents the pooled SD from D fits of multiple CCFs measured across the forward and backward directions within each channel, and the plotted D_eff is the mean across the two measurement distributions. c D_eff values from this figure (b) grouped into narrow (< 2.5 μm) and wide (> 4 μm) channel sizes (widths) based on the separation distance of ~3.6 μm (Sec EGFP: 7 narrow (5 emb), 5 wide (4 emb); Sqt-EGFP: 10 narrow (8 emb), 9 wide (9 emb); Sqt EGFP + Acvr2b Morpholino: 5 narrow (4 emb) 4 wide (2 emb); Sqt-EGFP + Lefty2: 6 narrow (5 emb), 5 wide (4 emb). The D_eff of Sqt-EGFP in narrow channels is significantly slower than in wide channels for all Sqt-EGFP and treatment cases (p < 0.0001), and Acvr2b Morpholino and Lefty2 speed up the slow Sqt-EGFP in narrow channels (p = 0.0007 and p = 0.0037, respectively). d Measurement of the transition of D_eff for Sqt-EGFP from D_loc (ACFs) across three distinct separation distances for different channel sizes (widths) (N_ch = 10, N_emb = 9). Average D and SD (shaded region) from grouped channels were plotted (N_ch = N_emb in brackets) with the error range being the pooled SD from individual channels, including forward and backward values. The D_eff slows from a higher D_loc value at distances larger than the channel size(width) (p = 0.0047 for comparison at 0 and 3.6 μm in narrow channels, p = 0.0751 for comparison at 0 and 5.2 μm in intermediate channels) while the D_eff remains relatively constant across space in wide channels (p > 0.3). Source data for the figures are provided in the Source Data file.

We also used these groups to measure the transition from D_loc to D_glob across the length of the channel (Fig. 5d). Following the receptor binding simulation results, in narrow channels (< 2.5 μm), D_eff dropped at a distance of 2 μm (p = 0.075) and remained low (p = 0.005 and 0.02 for D_loc and D_eff at 3.6, 5.2 μm) and relatively constant across space thereafter (p = 0.71 from 2 and 3.6 μm, p = 0.98 from 3.6 and 5.2 μm). For channel widths greater than 4 μm, D_eff changed little over space (p > 0.3). The channels with intermediate width had a similar D_loc with those of wide and narrow channels (p = 0.90 and 0.23, respectively). Similar to narrow channels, there was a slowdown at 2 μm (p = 0.0745) and D_eff remained constant thereafter (~30–31 μm²/s; p > 0.8 for all comparisons), with D_eff higher than those of the narrow channels but lower than those of wide channels (p = 0.1 and 0.02 respectively at 5.2 μm). Overall, these results recapitulate the simulation results and shows that the scale of the slowdown depends on the inter-cell spacing and that D_eff converges to different values depending on channel width.

Since Sqt is known to bind to Activin receptors, we investigated whether modulating receptor binding could influence D_eff over space. For this, we knocked down Acvr2b, a receptor known to bind to Sqt in vivo⁴⁸ using a morpholino⁵⁹ and measured D_eff at 3.6 μm to compare with the scatter plot of Sqt-EGFP’s diffusion. We found that D_eff in narrow spaces increased compared to native Sqt-EGFP values, with a wide distribution of observed D_eff values between 30 and 60 μm²/s (Fig. 5b). Grouping the narrow spaces together to compare with Sqt-EGFP showed a significant overall increase in the diffusion coefficient by about twofold (~18 to ~35 μm²/s; p = 0.0007), while the wide space D_eff increased slightly compared to Sqt-EGFP (~51 to ~58 μm²/s; p = 0.0742) (Fig. 5b, c). Furthermore, we measured the concentration gradient lengths for Sqt-EGFP across the blastula space and found that the knockdown of Acvr2b significantly increased the measured gradient length (Appendix G, Supplementary Fig. 8 and Supplementary Table 1; p < 0.0001).

Leftys are known inhibitors of Nodal morphogens. They act by either directly binding to the Nodal ligand or by competitively binding to receptors to prevent Nodal signalling^49,60,61. In zebrafish, Leftys inhibit both Squint and Cyclops⁶² and exhibit significantly higher diffusion coefficients compared to them across tissue spaces¹¹. We previously demonstrated that Squint binds to Lefty2 in the extracellular space⁴⁸. Here we examined whether overexpressing Lefty could speed up Squint’s diffusion since both direct interaction of Squint and Lefty and Leftys blocking available receptor binding sites should overall reduce the transient binding of Squint. Accordingly, we measured Sqt-EGFPs D_eff in the presence of Lefty by overexpressing Lefty2-mCherry in the extracellular space. Similar to the Acvr2b receptor morpholino result, we found a twofold increase in the diffusion coefficients for the Sqt-EGFP with Lefty2 overexpression in narrow spaces compared to native Sqt-EGFP values (~18 to ~38 μm²/s; p = 0.0037) while the D_eff in wide spaces remained relatively unchanged (~51 to ~51 μm²/s; p = 0.89, Fig. 5b, c).

It should be noted that in the transition from wide to narrow channels, the dimensionality reduction from 3D (wide channels) to 2D (narrow channels) leads to a reduction of the mean squared displacement from 6Dt to 4Dt^63,64. As our fit model cannot take account of this transition, this results in an underprediction of D by 30%. We tested this prediction using sec-EGFP and, barring a slow outlier in wide spaces (> 3 standard deviations slower than the mean), obtained apparent reductions in D on that order of magnitude for narrow channel spaces (p = 0.0004, p = 0.11 with outlier; Fig. 5b, c). Since this effect is also applicable to the diffusion of Sqt, the 30% underestimation of the diffusion coefficient in narrow spaces should result in a value of 35–40 μm²/s. However, the values measured for Sqt in narrow spaces are as low as 10 μm²/s and average around 18 μm²/s. A value of ~35–38 μm²/s is recovered only upon Acvr2b knockdown or Lefty2 overexpression.

Overall, these results suggest that modulating binding affinity does indeed influence D_eff of morphogens across space, and therefore, these morphogens experience hindered diffusion through binding, acting at the length scale of the intercell spacing.

The D_eff values measured at 3.6 μm across all wide channels (> 4 μm) for Sqt-EGFP (50–60 μm²/s) and Sec-EGFP (70–80 μm²/s) are very similar to the D_loc values obtained by previous confocal FCS measurements⁴⁸, demonstrating the equivalence between the Confocal FCS and SPIM-sFCCS fit results.

To further test the validity of our observations from Sqt-EGFP diffusion, we used SPIM-sFCCS to measure the D_eff of BMP2b-sfGFP across space. BMPs and Nodal signalling work synergistically to pattern the early zebrafish embryo ⁶⁵. Similar to Squint, BMP2b has also been shown to exhibit a difference in D_loc and D_glob values^12,66,67, and thus, can be useful to compare with Sqt results. From the measured D_eff values at 3.6 μm from a limited set of narrow (< 2.5 μm), intermediate width (2.5–4 μm) and wide channels > 4 μm (Appendix H, Supplementary Fig. 9 and Supplementary Table 2), we observed a distinct difference in diffusion dynamics between the channel types. We found that narrow channels significantly restricted ligand diffusion, slowing the molecules down by almost fourfold compared to that in wider channels in vivo (56–15 μm²/s; p = 0.0002), and the 2.5–4 μm channels yielded an intermediate D_eff value of ~31 μm²/s.

Discussion

Quantifying in vivo morphogen dynamics is crucial for understanding morphogenesis. However, current techniques, including FCS, SPT, and FRAP, probe different non-overlapping spatial scales and thus capture the spatial scale-dependent dynamics incompletely. To capture the full gamut of effects of tortuosity, extracellular matrix and receptor interactions over all length scales, therefore, requires techniques that can bridge the transition from local to global dynamics. Here, we employ SPIM-sFCCS with improved fitting models and analysis procedures to measure D_eff in vivo across intermediate-length scales.

We first used simulations to predict how a ligand’s D_eff changes over distance in inter-cell spaces with receptor interactions. For this purpose, we used single effective binding (p_bind) and unbinding (p_unbind) probabilities of a generic receptor as a qualitative model. We recognise that several receptors and co-receptors of different time-varying concentrations are involved, and their individual effect on receptor binding-based hindered diffusion can be hard to predict. Nevertheless, their overall influence can be captured within these generic simulation parameters, and the predictions should at least qualitatively capture the effect of the receptors’ interaction on morphogen dynamics.

Using this simplified model, receptor interactions can be tuned by either increasing the ligand-receptor affinity (binding and unbinding probabilities) or by changing the receptor concentration. Assuming a constant surface concentration of receptors on the cell membranes, the latter is equivalent to changing the spacing between the cell membranes that form the channels through which the morphogens diffuse. The simulations demonstrate that the interplay of geometrically constricted space, as well as reversible binding kinetics determined by the coupled parameters of binding and unbinding probabilities, together act to reduce the D_eff over space. These simulation results are consistent with those of Kuhn et al.²¹ and Zhu et al.¹⁵, who showed how binding affinity, receptor concentration, and tightness of space influence global diffusion. Furthermore, in addition to the findings of previous simulations, these simulations helped us predict that the transition from free to diffusion hindered by receptor interactions occurred at length scales on the order of the inter-membrane distance.

It should be noted that we make a distinction between the tightness of space as quantified by the width of the channel and the effects of the tissue’s tortuosity. Tortuosity measures the influence of obstacles that increase the path length for a particle to diffuse across two points in space. By excluding certain paths (mostly short, direct paths), the overall diffusion coefficient seems to slow down as molecules diffuse around cells at the global tissue scale. This can be quantified by FRAP measurements over scales larger than a cell diameter. But at the scale of the intermembrane spacing, which is one cell diameter or less, tissue tortuosity does not affect diffusivity.

To accurately measure D_eff over space, we had to derive the theoretical SPIM-FCS fitting models afresh. Due to the large detection volume per pixel and the extent of the light sheets, crosstalk between neighbouring pixels is strong and extends over a wide range. This was previously recognised, and solutions for deconvolution were developed⁶⁸. However, for SPIM-sFCCS, this leads to enhanced crosstalk that cannot be accounted for by previously derived analytical models⁴⁰. Correct D values can be extracted from these models but at the cost of an empirical calibration, which will overestimate the observation volume. This approach, however, fails for SPIM-sFCCS, and the derivation of a new model became necessary.

The disadvantage of this newly derived model is that no analytical solution can be found, and numerical integration is required to describe the fit function, which slows down data evaluation. At the same time, this makes the model versatile as any PSF model can be included in the numerical function, allowing the inclusion of even experimental PSFs⁶⁹. An alternative approach to data fitting is the evaluation by convolutional neural networks⁷⁰. Finally, we demonstrated that with the developed numerical fit model, we can extract correct Ds over distances ranging from at least 1–5 μm, spanning the accessible length scales between FCS and FRAP, and the range over which we expect the most drastic changes in D (Fig. 2b).

We then proceeded to apply SPIM-sFCCS in the living embryo. As the tissue within an embryo at 4 hpf can be dynamic⁵³, it was important to distinguish tissue movement artefacts, convection induced by tissue movement, and diffusion. In addition, the complexity of the 3D tissue topography and dynamics led to a wide distribution of D values in general. For the optimisation of SPIM-sFCCS in a highly dynamic tissue, we needed to find a compromise between limited measurement time and achievable SNR. As tissue moves fast in the early embryo and can corrupt the correlation functions, we could not measure over long times (mean measurement time ≈ 90 s). The resulting low SNR was mitigated by pixel binning and frame binning to obtain appropriate spatial cross-correlations and quantify D_eff. These strategies were crucial for the long-range measurements of the 3.6–5.2 μm separation distances shown.

Taking into account these limitations, we applied SPIM-sFCCS to the measurement of the scale-dependent diffusion coefficient of the nodal morphogen Sqt, dependent on receptor concentration and channel width. While the error of a SPIM-sFCCS measurement, depending on SNR, is about 20%, the in- and across-sample variability will increase the total error to about 40–50%. The sample variability within and across samples is due to two factors. First, the interstitial spaces in living systems are never exactly the same and will also slightly vary in their width even in the same channel-like spaces. In simulations, we show that subtle narrowing or expanding of the channel along its length increases or decreases D_eff (Supplementary Fig. 1e). This can add to errors in the D_eff measured in channels with non-uniform intermembrane distance across the length of the channel. Second, a developing embryo will always have some tissue dynamics. While we exclude any measurements where tissue dynamics dominate, we cannot completely exclude some minor effects of tissue dynamics in SPIM-sFCCS. Nevertheless, the difference between the measured diffusion coefficients in interstitial spaces for different channel widths, as well as over different distances, was statistically significant (Fig. 5c, d).

Our measurements supported the channel simulation predictions and showed that even after taking into account the effects of dimensionality reduction, Sqt slows at least twofold over the length of tight channels while remaining unperturbed in wide spaces. Both Acvr2b receptor knockdown and Lefty2 overexpression increased D_eff of Sqt in tight spaces while not affecting the diffusion in wide spaces, demonstrating the channel-width-dependent effect of receptor interactions. We recognise the presence of endogenous Sqt, which may competitively interact with receptors and endogenous Leftys that may bind to the expressed Sqt-EGFP. Both cases will cause an overestimation of D_eff. Nevertheless, the relative speedup of Sqt-EGFP in the presence of Lefty2 overexpression and Acvr2b knockdown in comparable tight spaces was still significant. Furthermore, SPIM-sFCCS measurements on BMP-2b also showed differences in the D_eff measured in narrow and wide spaces, thus suggesting the general applicability of the mechanism of space-dependent binding-hindered diffusion for morphogens.

A more detailed analysis of the experimental D values of Sqt in channels of different widths showed a gradual decrease of D over distance, which depended on the channel width, as predicted by the simulations. To our knowledge, this is the first simultaneous measurement of diffusion coefficients in vivo across spatial scales, from 0 to 5.2 μm, that captures the gradual slowdown of morphogen diffusion, and thus demonstrates the power of SPIM-sFCCS.

These D_eff measurement results are consistent with previously reported receptor binding hindered diffusion of ligand measurements in vivo^{13,15,17,18,21}. Previous measurements by ref. ²¹ using SPT present three distinct diffusive populations in their cell–cell interface spaces, a very low D suggesting an immobile population (~0.5 μm²/s), an intermediate value (1.2–3 μm²/s) suggesting a membrane-bound diffusion fraction and a fast-diffusing fraction (16–17 μm²/s). Our D_eff results of Squint-EGFP in narrow channels ~18 μm²/s are in good accordance with the fast diffusing fraction. Unfortunately, immobile components cannot be captured by FCS, and as shown in Fig. 3d and Supplementary Fig. 5d, the spatial range for accurately measuring slow D values (1–10 μm²/s) is very low and will require significantly higher measurement times, which is difficult to manage in a growing-developing tissue. However, fitting the autocorrelations with a two-component fitting model can capture the local extracellular fast component and the slow component near the membranes and has been used in past morphogen FCS measurements (refs. ^17,48,71,72). Here, we limit our results to only capturing the fast-diffusing component, since we consider that to be the one diffusing and slowing down across the tissue and acting as the dominant factor for long-range signalling.

Our previous study using two-colour FCCS⁴⁸ revealed a high Acvr2b-Sqt affinity with a dissociation constant K_d of 60 nM and, consistent with this, our results from knocking down Acvr2b resulted in a speed-up of Sqt diffusion. Our results are also consistent with previous receptor knockdown observations showing increases in D_glob from knocking down Acvr1b-a/b or in the absence of co-receptor Oep¹³. In contrast, however, it was shown that Acvr2b receptor overexpression resulted in an insignificant overall decrease in D_glob¹³, and titrations of Oep expression increased the fraction of bound molecules but did not change D_loc²¹. Overexpression of Oep did, however, decrease D_glob as measured by FRAP¹³, probably due to the increased ligand capture by the potent Oep co-receptor influencing its global mobility. The speed-up in ligand diffusion upon receptor knockdown but minimal change due to receptor overexpression suggests the possibility of a ligand-limited receptor binding system. Excess receptors from overexpression may not necessarily increase the binding rate, whereas knockdown of receptors reduces the binding-based hindrance experienced by the ligands. The presence of ubiquitous expression of Oep, Acvr1b and Acvr2b receptors across the early blastula¹³ and later enriched Oep and Acvr1b-a expression in the embryonic margins (refs. ^13,73), where Nodals are active at shield stage, in addition to uniform Acvr2b blastula expression, supports the ligand-limited model. Moreover, Oep replenishment to counteract receptor saturation⁵⁰ also helps in maintaining high receptor levels. Future experiments quantifying ligand binding kinetics with other receptors and receptor occupancy will be required to confirm the ligand-limited binding mechanism.

Our group also previously demonstrated receptor binding hindered diffusion in tight interstitial spaces of the 24–48 hpf zebrafish cerebellum for Wnt3, where knocking down co-receptor LRP5 increased D_glob values¹⁷. Furthermore, it was shown that BMP4 gradients in mice formed due to an interplay of geometry-constricted ligand diffusion in narrow interstitial spaces and restricted basolateral localisation of receptors⁷⁴, thus highlighting the importance of the coupled effects of tissue geometry and receptor binding considerations for morphogen dispersal.

Our results also lead to further predictions. Endogenous nodal signalling is highly active at the margin regions of the blastula⁷⁵. As these margins still have intact cell–cell adhesions⁷⁶, the inter-membrane spaces in such regions are below 100 nm and, therefore, below the diffraction limit. Such tight spaces should cause a significantly greater slowdown for ligands. Due to the diffraction limit, we cannot quantify the tight spaces. Nevertheless, based on our results, we extrapolate that D_eff values in these denser tissue regions will be much lower than our results for 2 μm channels of 10–20 μm²/s.

Based on our scale-dependent D_eff results, we illustrate a plausible mechanism by which molecules slow down from local to global length scales in Fig. 6. Our measurements of Sqt’s D_eff transition across space suggest that the receptor binding-based slowdown from the molecular weight-limited diffusion regime to the binding-hindered diffusion regime happens within the interstitial space between a pair of cell membranes (Fig. 6a). Beyond the intermembrane spacing distance, the resultant D_eff value is maintained across the length of the interstitial space. After transversing the intermembrane space, the molecules experience the tortuosity of the tissue (Fig. 6a right). They must circumvent cells, which delays their dispersal and causes a further slowdown. We predict that the transition from a fast D_loc to a D_eff reduced by receptor binding, and if applicable by interactions with extracellular matrix components, occurs within one cell diameter itself. The final D_glob beyond the scale of a single cell, which helps shape the concentration gradient, is then determined by tissue tortuosity (Fig. 6b). In wider tissue spaces, the initial slowdown due to receptor binding is not significant. However, the effect of tissue-wide tortuosity proportionally reduces the D_eff, resulting in a slowdown but a relatively higher value compared to tightly packed tissues (Fig. 6b). Since tissues typically have varying intercell spacings, the overall effective diffusion coefficient will be a weighted average of the diffusion coefficient over all the different channel widths. Potentially, this can explain how Nodals’ and other morphogens’ mobility is regulated in different tissue environments in vivo.

Fig. 6. Plausible mechanism of morphogen slow down across space for the formation of concentration gradients.

a Illustrations to show how interactions at different length scales influence the diffusion of morphogens. At the scale within the inter-membrane space, morphogens exhibit free diffusion or ECM interaction-hindered diffusion. Beyond the length scale of the inter-membrane spacing and within the channel-like interstitial space, morphogens experience the influence of receptor binding. Beyond the intermembrane space, the tortuosity of the tissue requires the molecules to diffuse along a longer route around the cell’s perimeter, further slowing molecules down. b An illustrative plot for the transition from a local fast to a global slow diffusion coefficient across space. In narrow spaces, the initial slowdown is significant due to a higher propensity to bind to membrane receptors compared to wide spaces. Beyond the channel length, the influence of tortuosity is similar for both cases, and the final D_eff value is reached within the scale of one cell diameter. Created in BioRender. Nelanuthala, A. (2026) https://BioRender.com/4zm1p4s.

While our work focused exclusively on investigating Sqt diffusion and the effects of the receptor Acvr2b and the inhibitor Lefty2, we also recognise that the influence of Lefty1 and dynamics of Cyclops and its interactions with the receptors and inhibitors are required for the complete understanding of Nodal dynamics in vivo.

Previous engineered gradient studies show that receptor-based binding is sufficient to form morphogen gradients (refs. ^19,20). Nodal gradients were also previously shown to be shaped by receptor binding¹³ and co-receptor Oep’s preferential localisation in the margins⁷³ and maintained by receptor replenishment⁵⁰. Our finding that knocking down Acvr2b increases Sqt’s gradient length is in line with and adds to the previous observations of Sqt’s gradient length increase in the absence of Acvr1b and Oep. However, it must be noted that developmental phenotype defects at early embryo stages were previously not observed for the single knockdown of Acvr2b-a (refs. ^13,59) but only for the double knockdown of both Acvr2b receptors (-a and -b)¹³, suggesting a possibility of compensatory mechanisms to regulate signalling similar to Lefty1 and 2’s mutual compensatory mechanism⁶².

Mesoendoderm patterning, a consequence of nodal signalling, remains robust even when interstitial spaces in the blastoderm are perturbed by drug-induced fluidisation or compaction⁷⁷. This suggests that inter-cell spacing and the resulting tissue packing, standalone, is unlikely to fully dictate nodal gradient formation. Robust patterning can be brought about, in part, by compensatory buffering from the nodal inhibitor Lefty, which was previously shown to help restrict the long-range nodal spread⁶² and scale nodal signalling⁷⁸. Furthermore, nodal signalling is not only determined by the concentration gradient but also by the duration of nodal exposure (refs. ^52,79). The blastoderm’s margin regions, where nodal signalling is active, have intact cell–cell contacts compared to the central blastoderm due to localised non-canonical Wnt11 signalling at the margins⁷⁶. It was shown that increased cell–cell adhesions increased nodal signalling, which in turn further strengthened the cell contacts⁷⁹. The differences in strengths of cell–cell contacts and the resulting nodal signalling resulted in differential cell fates⁷⁹.

Our observation that Squint slows down rapidly within the scale of the channel’s width suggests a physical mechanism that not only helps with restricting nodal dispersal but also allows for increased nodal signalling within the tight cell junctions in the margins, where nodal signalling is active. In line with this interpretation, a recent article proposes a nodal-tissue rigidity feedback loop, where nodal signalling-driven tissue rigidification further causes a restriction in the length-scale of nodal signalling⁸⁰.

Leftys, which are pre-repressed and expressed at a later time⁵¹, allow for a time window for nodal signalling. It was previously shown that Lefty needs to diffuse rapidly to adequately inhibit nodal dispersal¹¹. Our results showing Sqt speed up in the presence of Lefty2 suggest the possibility of rapid clearance of nodals to effectively regulate nodal signalling.

Initial measurements of morphogen dispersal in zebrafish suggested that Sqt followed the reaction-diffusion-based patterning system with its inhibitor Lefty¹¹. However, in vivo diffusion measurements of morphogens also presented a confounding observation regarding the presence of a fast local and slow global D value (Yu S.R. et al. ⁴ for Fgf8, Muller P. et al. ⁵ for Sqt, Cyc and Fgf8) with the suggested cause for the slowdown being receptor binding-based trapping⁵. Follow-up studies reliably quantified Sqt’s diffusion in vivo (refs. ^16,48) and its interaction strength with receptors and inhibitors⁴⁸. Subsequent studies also documented the changes in Sqt’s diffusion across different interstitial spaces in the tissue²¹ and due to its interactions with its receptors Oep and Acvr-1b¹³, and, implicatively, also changes to its gradient spread (refs. ^13,50), adding more evidence for the receptor-hindered diffusion model.

Our SPIM-sFCCS-based results here not only confirm the combined influence of receptor interaction and tissue geometry on morphogen diffusion and add Acvr2b-a and Lefty2 to the list of Sqt’s diffusion regulators but, more importantly, the simultaneous spatial D_eff measurement uncovers the length scale at which the slowdown occurs. This allows us to infer details about the hindered diffusion model that were previously not accessible and provides a stronger foundation for the model, and helps bridge the local to global diffusion transition gap.

The implementation of SPIM-sFCCS provides a useful tool for investigating the diffusion coefficients in live embryos in great detail. It achieves access to the intermediate range of up to at least 5 μm, which is not accessible by more traditional measurement techniques. Our data from both Sqt and BMP-2b indicates a slowdown of morphogen diffusion with distance within the extracellular space that is dependent on reversible membrane binding and intercell spacing, leading to an effective reduced D_glob already at a distance comparable to the diameter of a cell. We expect that with further technical developments, detailed maps of diffusion coefficients can be measured in live embryos, providing a much more detailed picture of morphogen transport than is available today.

Methods

Animal studies ethics approval

All zebrafish-related experiments were performed following the protocols IACUC #BR18-1023 or IACUC #BR22-0823 approved by the Institutional Animal Care and Use Committee at the National University of Singapore. All zebrafish used in this study belonged to the National University of Singapore, Department of Biological Sciences wild-type strain. The embryos were obtained from the Comparative Medicine vivarium or the Department of Biological Sciences fish facility at the National University of Singapore. All experiments were performed within 0–5 h post fertilisation (hpf), which did not allow for the identification of the sex of embryos during experiments.

Simulations for diffusion and binding in narrow channels

The diffusion of morphogens in a developing embryo is hindered not just by the geometry of the tissue but also by binding and unbinding to receptors on the cell walls. We simulated a simplified version of this situation in 2D by creating a channel of width d_width bounded by two parallel lines parallel to the x-axis, representing cell borders. All particles started diffusing from the centre point of the channel. The simulation time step was set to Δt = 0.01 ms, and the diffusion coefficient was D = 60 μm²/s. The time step of 0.01 ms ensures a short step size of $\sqrt{4Dt}$ = 0.049 μm for 60 μm²/s so that particles will require multiple steps to cross the channel. Particles perform a random walk within the channel with step size Δr in x and y direction, drawn from a Gaussian distribution (mean 0, variance 2DΔt). Once a particle hits one of the walls, it can either bind with probability p_bind or be reflected as determined by drawing a number from a uniform distribution between 0 and 1 and comparing it to p_bind. A bound particle has a probability p_unbind to unbind and continue diffusion in the next simulation step, determined by drawing a number from a uniform distribution between 0 and 1 and comparing it to p_unbind. As all particles start at the centre point of the channel, the distribution of the particles along the x-axis, that is, along the channel, provides a measure of the effective diffusion coefficient (D_eff) resulting from the free diffusion and transient binding-unbinding dynamics over space. The distance that particles traversed along the x-axis for each time interval (1 ms) was quantified using the RMSD, which was calculated from the variance of the resultant distribution of particle positions x as $\sqrt{⟨x^2⟩}$. D_eff was also determined along the length of the channel at time t as

$$D_{eff}=\frac{⟨x^2⟩}{2t}$$ 1

The number of particles in each simulation was scaled according to the width of the channel, as 4000 particles/μm of channel width. This ensured that there were sufficient particles to create a Gaussian distribution whose variance could be calculated appropriately. For simulations of channels of different converging or diverging shapes (Supplementary Fig. 1e), channel convergence/divergence was set to 16% (or 32% for 2× slope) of their width over 10 μm. All simulations were performed in Mathematica version 12.3 (Wolfram Research, Illinois, USA).

SPIM setup

The light sheet microscope system and acquisition settings were based on the details presented in ref. ²⁶ and are briefly described here. A 488 nm laser (Cobolt 06-MLD 488nm 0488-06-01-0100-100, Cobolt AB, Sweden), required to excite EGFP, was co-aligned with a 561 nm laser (Cobolt 06-DPL 561nm 100 mW 0561-06-91-0100-100, Cobolt AB, Sweden), used to excite mApple/mCherry. The beams were directed into a single-mode optical fibre (kineFLEX-P-3-S-405.640-1.0-4.0-P2, Qioptiq, USA), which expanded the beams by 1.3 times. The output beams were passed through a beam expansion (f₁ = 30 mm AC254-030-A, f₂ = 60 mm AC254-060-A; Thorlabs Inc., USA) to obtain a further 2× expansion, which was required to fill the back aperture of the illumination objective. The beams were then passed through an achromatic cylindrical lens of 75 mm focal length (ACY254-075-A; Thorlabs Inc., USA), followed by an illumination objective (SLMPLN 20×/0.25; Olympus, Tokyo, Japan). The light sheet thickness obtained for the 488 nm beam had a 1/e² radius of approximately 1.1 μm and was maintained for all SPIM-sFCCS measurements. The light sheet waist was aligned to coincide with the optical axis of the detection objective (LUMPLFLN 60×/1.0, Olympus, Tokyo, Japan). The detection objective was placed in a mounting hole on one side of a custom-made sample chamber and mounted on a piezo objective scanner (P-721 PIFOC; Physik Instruments, Germany) to control the objective with nanometer precision. The sample was mounted on a motorised stage with three linear positioning systems (Q-545 Q Motion Precision Linear Stage; Physik Instruments, Germany) with piezo motors for the three axes and one rotation stage (DT-34 Miniature Rotation Stage; Physik Instruments, Germany). The fluorescence emission detected by the objective was passed through a bandpass filter (FF03-510/20-25, Semrock, USA) to capture EGFP emission and FCS, or a dual-band bandpass filter (FF01-512/630-25, Semrock, USA) to assess mApple/mCherry with EGFP co-expression, and was projected onto an EMCCD camera (Andor iXon3 860, 128 × 128 pixels; Andor Technology, Belfast, UK) by a tube lens (LU074700, f = 180 mm, Olympus, Tokyo, Japan). A flip mirror was used to alternatively project the image onto a sCMOS camera (OCRA-Flash4.0 V2; C11440; Hamamatsu, Shizuoka, Japan) with a large field of view to visualise the global position of the embryo and to decide on the region of interest.

Time series stacks were obtained and saved in TIFF format from the EMCCD by the software Andor Solis for Imaging (version 4.31.3037.0), and were subjected to SPIM-sFCCS analysis depending on the acquisition parameters of exposure and measurement time used.

Free diffusion simulations for SPIM-sFCCS

Simulations of free diffusion in SPIM (3D) were performed in Imaging FCS 1.52⁸¹, an ImageJ (http://imagej.nih.gov) plugin building on the 2D TIRF simulations described in Sankaran et al. (2013)⁸² using the simulation parameters given below. Briefly, we assumed that fluorescence signals were collected through a 60×, NA 1.0 microscope objective for SPIM. The emitted signal was imaged onto a camera with a pixel size of 24 μm (object space pixel size = 400 nm), and of which a 20 × 20 pixel area is observed (corresponding to 8 × 8 μm² in SPIM). To ensure freely fluctuating particle numbers in the observed area/volume, the simulation space was chosen to be 3 times larger in x and y dimensions than the observed area and additionally 20 times larger in z direction than the light sheet thickness (1/e² radius) with the observation volume centred in the simulation space. A defined number of particles was distributed randomly in space. For every simulation step of time Δt, particles were displaced in each of the three dimensions by a random step size drawn from a Gaussian distribution with mean 0 and variance 2DΔt, where D is the diffusion coefficient. If a particle left the simulation space, a new particle was created randomly on the border of the space. The time step Δt was chosen to be 1/10th of the exposure time T of the camera to account for particle movement during image acquisition. At each simulation step, each particle emitted a number of photons sampled from a Poisson distribution with a mean of F(z)cpsΔt, where cps represents the molecular brightness in counts per particle per second and F(z) is a scaling factor taking into account the axial dependence of the excitation intensity. The emitted photons were distributed on the camera with a probability distribution as defined by the point spread function (PSF) of the microscope objective in the xy plane. The PSF was approximated as a Gaussian function with a 1/e² radius of

$${\omega }_{xy}={\omega }_0\lambda /NA$$ 2

Here ω₀, a constant, was obtained from a calibration experiment by ensuring that the diffusion coefficient of a dye in an aqueous solution is constant independent of binning (refs. ^26,83) and was found to be equal to 1. The scaling factor F(z) is given by a Gaussian distribution of the light sheet intensity in the z direction.

$$F\left(z\right)=\frac{1}{\sqrt{\frac{\pi }{2}}{\omega }_z}\left(e^{\frac{-2z^2}{{\omega }_z^2}}\right)$$ 3

where ω_z is the 1/e² value of the light sheet’s thickness. As particles outside the focal plane will result in a wider distribution of the photons on the camera, photons were spread over a PSF with

$${\omega }_{xy}\left[z\right]={\omega }_{xy}+\mid z\mid \tan \left(\alpha \right)$$ 4

The detected counts were summed up and projected onto a 2D image every 10 simulation time steps (T = 10Δt), representing one frame in the image stack. A maximum of 500,000 frames was simulated for SPIM-sFCCS analysis. To simulate the effect of camera noise, spurious counts were randomly drawn from a Gaussian distribution (mean 0, standard deviation σ = N_noise) and added to each pixel in each frame.

For simulation results shown in Figs. 3b–d and 4b, c, e and Supplementary Fig. 5, image stacks were simulated and analyzed with 5000 particles; 100,000 frames; exposure time = 1 ms; 30,000 cps; ω_xy = 515 nm; ω_z/light sheet thickness = 1.13 μm; 1 × 1 binning unless mentioned otherwise for the specific sub-figure.

Fluorescent bead measurements procedure

Diffusion measurements of fluorescent beads in solution were performed based on the procedure described previously²⁶. Briefly, fluorescent beads (FluoSpheres, Invitrogen) with a 100 nm diameter and emitting fluorescence signal at 515 nm wavelength were diluted 100× of the stock solution in water and used here for Imaging-FCS measurements. The bead solution was sonicated for 30 min prior to measurement. 50 μl of the bead solution was enclosed in heat-sealed FEP (fluorinated polyethene-propylene) bags with a film thickness of 25 μm (S1833-04, 1000 × 150 × 0.025 mm FEP foil, Bola (Bohlender), Germany). The sealed bag was mounted into the sample chamber, and the signal from the fluorescent beads was captured for SPIM-sFCCS analysis. 80,000 frames with an exposure of 2 ms for a laser intensity of 12 W/cm² were acquired for the measurements.

mRNA preparation, injection, zebrafish mounting and SPIM imaging

The Sqt-EGFP, sec-EGFP and the Lefty2-mCherry plasmid constructs used here were previously published by ref. ⁴⁸. The plasmids were linearised using the NotI restriction endonuclease (New England Biolabs, USA). mRNA was transcribed from the plasmid using a transcription kit (AM1340 mMESSAGE mMACHINE SP6 Transcription Kit, Thermofisher Scientific, USA) and purified using Dynabeads mRNA Purification Kit (Thermofisher Scientific, USA). The morpholino against Acvr2b (GeneTools, USA) is based on the antisense sequence (GCAGAGAAGCGAACATATTCCTTT) in ref. ⁵⁹. The paralog for Acvr2b morpholino used here is Acvr2b-a and was identified previously¹³.

10–15 pg of Squint-EGFP mRNA or sec-EGFP mRNA were injected into a single cell of the eight-cell stage embryo for SPIM-FCS measurements (Supplementary Fig. 6a top left). This concentration is higher than 5–10 pg used for confocal FCS previously⁴⁸ to ensure that sufficient signal was captured through the SPIM system and the EMCCD camera. Higher concentrations may increase SNR for imaging but make it harder for the fluctuations in intensity with respect to the background intensity to be appropriately detected, which is crucial for FCS.

For Morpholino injection, a cocktail of the Morpholino and PMT (plasma membrane targeting sequence)-mApple mRNA was injected in each of the cells of the two-cell stage to ensure that only fertilised embryos were injected. The concentration of the injected morpholino was 2.5 μg/μl. This is in the same order 4 μg/μl used to knock down the expression of Acvr2b in the report by ref. ⁵⁹. Along with it, 50 pg of PMT-mApple mRNA⁸⁴ was injected to visualise where the injected cocktail with morpholino acted in the embryo and ensure that the cell membranes flanking the regions measured had their Acvr2b receptors knocked down.

Lefty2-mCherry mRNA was similarly injected in each of the cells of the two-cell stage in the embryo, resulting in a total injection load of ~60 pg to ensure adequate over-expression across the extracellular space.

The Bmp2b-sfGFP construct used here was previously published by ref. ¹². The plasmid was linearised using the NotI restriction endonuclease (New England Biolabs, USA). mRNA was transcribed from the plasmid using a transcription kit (AM1340 mMESSAGE mMACHINE SP6 Transcription Kit, Thermofisher Scientific, USA) and purified using the Zymoclean RNA Clean & Concentrator kit (Zymo Research, USA).

Similar to Sqt-EGFP mRNA microinjection, 10–15 pg of Bmp2b-sfGFP mRNA was injected into the single cell of an eight-cell stage embryo for SPIM-FCS measurements.

The embryos were screened at 3 hpf for EGFP/sfGFP/mApple/mCherry expression in a fluorescence stereo microscope (Supplementary Fig. 6a top right). For SPIM-FCS measurements, the selected embryos were dechorionated and suspended in 1% low-melting agarose (UltraPure™ Low Melting Point Agarose, 16520100, Thermofisher Scientific, USA) in a Fluorinated Ethylene Propylene (FEP) tube of 1.1 × 1.5 mm² cross-section (FT 1.1 × 1.5, Adtech Polymer Engineering, England) (Supplementary Fig. 6a bottom left). The embryos were mounted and imaged between 3.5 and 5 hpf before severe cell movements made FCS analysis difficult.

To minimise imaging artefacts, the embryo was oriented in a way that the blastula space, based on the region of interest for imaging, was oriented to be closest to one face, and the animal pole of the embryo was placed as close to the wall of the FEP tube. Essentially, the embryo’s animal pole was positioned within one quadrant of the cylindrical tube (Supplementary Fig. 6a bottom left). This ensured that the light sheet entering the tube could illuminate a plane of the embryo’s tissue optimally and that the detection objective could capture the fluorescence emission without any losses during the propagation through the tube and agarose.

Initially, the embryo was imaged on the sCMOS to visualise the embryo orientation and choose a broad region of interest (221.9 × 221.9 μm) (Supplementary Fig. 6a bottom middle). Once chosen, we switched to the EMCCD camera to image a smaller ROI (51.2 × 51.2 μm) (Supplementary Fig. 6a bottom right). Imaging with the sCMOS camera also allowed us to visualise the global spread of Lefty2-mCherry and Acvr2b morpholino (in cocktail with PMT-mApple) prior to the FCS measurements (Supplementary Fig. 6b, c). To find appropriate channels like inter-membrane spaces, the tissue space was surveyed with low laser intensities (30–60 W/cm²) and high exposures (50–100 ms) to reduce photo-bleaching. We avoided ROIs showing cell movements, shadows, or scattering artefacts. Once the ROI was chosen for SPIM-sFCCS and the field of view was cropped to allow for high frame rates, image stacks with about 50–200 k frames were recorded with low exposures (0.4–2 ms) and high laser intensities (60–600 W/cm²). Since we ensured low expression concentration and chose high frame rates that result in high read noise, we chose to have high laser intensities to ensure that the fluctuation intensity was maximised for FCS, without significantly photobleaching the fast diffusing molecules seen in this study. Within the margins of error, the intensity in this range did not change the measured D⁸³.

SPIM-sFCCS analysis

Autocorrelations and spatial cross-correlations on the acquired and simulated image stacks were performed using an ImageJ plugin: Imaging FCS 1.52 modified for the fitting model given below. The intensity traces of the different pixels were autocorrelated, or traces from two specific pixels separated by a certain distance in x and y directions were cross-correlated over time τ as given by the following equation.

$$G\left(\tau \right)=\frac{⟨F\left(t,x,y\right)F\left(t+\tau ,x+\delta x,y+\delta y\right)⟩}{⟨F\left(t,x,y\right)⟩⟨F\left(t+\tau ,x+\delta x,y+\delta y\right)⟩}$$ 5

In the case of autocorrelations, δx and δy are 0. The diffusion coefficients were obtained by fitting the auto- and cross-correlation curves to a free 3D diffusion fitting model. The fitting equation derived in ref. ⁴⁰ was modified to account for crosstalk between pixels from out-of-focus particles (Eq. (4)). This modifies the detection volume, which is illustrated in Fig. 3a. The modified cross-correlation model was thus

$$G\left(\tau \right)=\frac{1}{4{\pi }^{3/2}{a}^4\sqrt{D\tau }⟨C⟩{\omega }_z^2}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{e}^{-\frac{{\left(z-z^{\prime }\right)}^2}{4D\tau }-\frac{2z^2}{{\omega }_z^2}-\frac{2z^{\prime 2}}{{\omega }_z^2}}$$

$$\left[\frac{A\left(z,z^{\prime }\right)}{\sqrt{\pi }}\left(e^{-\frac{{\left(r_x+a\right)}^2}{A{\left(z,z^{\prime }\right)}^2}}+e^{-\frac{{\left(r_x-a\right)}^2}{A{\left(z,z^{\prime }\right)}^2}}-2e^{-\frac{r_x^2}{A{\left(z,z^{\prime }\right)}^2}}\right)\right)$$

$$\left(+\left(r_x-a\right)\mathrm{Erf}\left(\frac{r_x-a}{A\left(z,z^{\prime }\right)}\right)+\left(r_x+a\right)\mathrm{Erf}\left(\frac{r_x+a}{A\left(z,z^{\prime }\right)}\right)-2r_x\mathrm{Erf}\left(\frac{r_x}{A\left(z,z^{\prime }\right)}\right)\right]$$

$$\left[\frac{A\left(z,z^{\prime }\right)}{\sqrt{\pi }}\left(e^{-\frac{{\left(r_y+a\right)}^2}{A{\left(z,z^{\prime }\right)}^2}}+e^{-\frac{{\left(r_y-a\right)}^2}{A{\left(z,z^{\prime }\right)}^2}}-2e^{-\frac{r_y^2}{A{\left(z,z^{\prime }\right)}^2}}\right)\right)$$

$$\left(+\left(r_y-a\right)\mathrm{Erf}\left(\frac{r_y-a}{A\left(z,z^{\prime }\right)}\right)+\left(r_y+a\right)\mathrm{Erf}\left(\frac{r_y+a}{A\left(z,z^{\prime }\right)}\right)-2r_y\mathrm{Erf}\left(\frac{r_y}{A\left(z,z^{\prime }\right)}\right)\right]dzd{z}^{\prime }$$ 6

where $A\left(z,z^{\prime }\right)=\sqrt{\frac{1}{2}\left(8D\tau +{\omega }_{xy}{\left[z^{\prime }\right]}^2+{\omega }_{xy}{\left[z\right]}^2\right)}$ a = side length of a square pixel D = the diffusion coefficient ω_z = the 1/e² radius of the light sheet thickness r_x and r_y are distances between pixel pairs in x and y for which the cross-correlations are performed.

〈C〉 = time averaged concentration; C is related to the number density N in the pixel observation volume V by 〈C〉 = 〈N〉/V.

This equation was numerically integrated between −2ω_z and 2ω_z to capture almost all of the information for the Gaussian beam in the z-axis. The derivation for the above equation can be found in Appendix C.

The value for the pixel size (a) used for fitting was 400 nm, the value ω_xy used was 515 nm, and the value of ω_z was 1.13 μm.

The recorded data was checked for any movement and drift-related artefacts as described in Optimising SPIM-sFCCS for in vivo measurements section by checking the forward and reverse fit results and including only the results where the difference between the values was within 30% of the mean D value.

The limited time resolution of the camera (~1 ms) can result in the inaccurate fitting of auto-correlations capturing fast diffusion coefficients (60 μm²/s in Fig. 3d). Moreover, unlike in simulations, subtle background cell movements can cause low-frequency fluctuations in fluorescence intensity traces of the acquired data, which are unrelated to diffusion. These artefacts influence the longer lag time correlations, often resulting in correlation functions similar to a two-component ACF function, or can bias the single component fits to lower D values. Therefore, to reliably fit autocorrelations from acquired data, we binned pixels to create a large area for molecules to spend sufficient time to provide statistics for autocorrelations and only included the early lag times (< 100 ms) for fitting (Fig. 3e and for results in Fig. 5d).

All CCFs were fit assuming the presence of only a single diffusing component. In the fluorescent bead diffusion analysis, the ROI was chosen to reflect the lowest fit N (conc.) and highest D in the solution to ensure the thinnest volume from the light sheet²⁶ was used and to avoid including any intensity spikes from bead clusters, which cause artefacts in the correlations. For analysis of in vivo measurements, the ROIs were chosen to ensure that the channel-like space with a constant width was captured while also ensuring that drift-related artefacts did not bias the results. Channel widths were quantified by measuring the full width at half maximum across the thinnest region of the channel with a constant width. Since pixel binning of at most 4 pixels was used, the CCFs over space were taken apart with 5 (2 μm), 9 (3.6 μm), 13 (5.2 μm) pixel separation distances chosen (Fig. 5d)

For diagonal channels (examples in Figs. 4f and 5a narrow channel, Supplementary Figs. 7c and 9a, b), a combination of x and y separations resulting in the equivalent separation distances as given above was used.

Statistics and reproducibility

Simulation results for diffusion in channel-like spaces in Mathematica (Fig. 2c, Supplementary Fig. 1c, d) and SPIM-sFCCS (Fig. 3c and Supplementary Fig. 5b, d, e, f) were obtained from the evaluation of 3 independent simulation seeds. For different experimental treatment cases, while classifying channel width-based results into narrow and wide cases, the results were replicated through independent channel measurements across multiple embryos, as given in the sample sizes for Fig. 5c, d and Appendix H Supplementary Table 2 in their respective captions. Similarly, for the Sqt EGFP gradient measurements in Appendix G Supplementary Table 1, results were replicated through independent measurements across multiple embryos for each of the Acvr2b morpholino-treated and untreated cases.

Two-sided Welch’s t-tests were performed in Origin Pro 2026 (OriginLab Corporation, Northampton, MA, USA) to quantify pairwise differences across different channel types in Fig. 5c, d, as well as across different separation distances in Fig. 5d, and to assess the influence of Lefty2 addition or Acvr2b knockdown in Fig. 5c.

No statistical method was used to predetermine sample size. For SPIM-sFCCS, from the measured image stacks, ROIs containing channel-like space were cropped, and only the areas within the channel were included for evaluation. As described in the Optimising SPIM-sFCCS for in vivo measurements (Results) and SPIM-sFCCS analysis (“Methods”) sections, measurements exhibiting substantial cell movement (30% of measured data) or in which D_fwd and D_bwd differed by more than 30% from the mean D were excluded (40% of measured data). After this filtering step, no additional data were excluded from the analysis.

Reporting summary

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

Author contributions

T.W., K.S. and T.J.C. conceived the project. A.V.S. and B.S. performed SPIM-FCS measurements. A.V.S. and J.S. analysed the sFCCS data. C.T. performed gradient length measurements and analysed its data. A.V.S. wrote the first draft, and all authors edited the final manuscript.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

T.W. and T.J.C. acknowledge funding from the Singapore Ministry of Education (MOE2016-T3-1-005). K.S. acknowledges funding from UKRI-BBSRC and Leverhulme trust. A.V.S. and B.S. are supported by NUS Research Scholarships.

Data availability

The data generated through simulations, raw data files for SPIM-sFCCS measurements and morphogen gradient measurements, along with the analysis results generated in this study, have been deposited in the NUS ScholarBank database under accession code https://scholarbank.nus.edu.sg/handle/10635/318321The figure and table data generated in this study are provided in the Source Data file with this paper. Source data are provided with this paper.

Materials availability

Plasmids and maps can be provided on request.

Code availability

ImFCS is a plugin in Fiji/ImageJ and is available publicly through Fiji or at the following url: https://github.com/Biophysical-Fluorescence-Laboratory/Imaging_FCS/tree/v2_0_4. The Mathematica code is available at the following url: https://zenodo.org/records/18998191.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-73639-3.