Nematic cell alignment directs calcium waves in an epithelial monolayer

Abstract

Tissues rely on supracellular signals to coordinate their cells over a long range. Two such tissue-scale cues are calcium waves and patterns of cell-cell alignment or nematic order. During wound healing, for example, calcium waves propagate across a tissue to guide directed cell migration and reepithelialization. Defects in long-range cell-cell alignment, or nematic orientation, can act to localize morphogenetic events in a tissue. Although these two cues have been considered in isolation, we demonstrate a relationship in epithelial tissue between long-range calcium signaling and the cell’s nematic order: The speed of a wound-induced calcium wave depends on the angle between the wave vector and cell axis, with maximal wave speed occurring perpendicular to the tissue’s orientation. Including anisotropic diffusive coupling between cells in a canonical reaction-diffusion model recapitulates our measured calcium wave dynamics. Our model demonstrates how orientation defects could desynchronize information propagation across a tissue. The anisotropic diffusivity in our model leads to a calcium wave front being bent around nematic defects. Therefore, cells the same distance from a wound can receive the calcium signal at different times. Our work elucidates how spatial patterns in global cell alignment can control collective communication via calcium signaling during development, wound healing, and disease.

Significance

Cells exploit global cues to collectively coordinate their behaviors inside tissues. Both calcium waves and long-range cell orientation patterns can relay positional information to cells in a tissue, guiding localized cell behaviors. We show that in epithelial tissue of elongated cells, there is an interaction between these two means of relaying information. The speed at which calcium waves move across a tissue is modulated by patterns of cell-cell alignment. Therefore, cells at the same physical distance from a stimulated wave receive the signal at different times. Our work sheds light on how long-range calcium signals present, for example, during wound healing, embryo development, and cell extrusion are guided by patterns of cellular alignment.

Introduction

Cells rely on supracellular cues to collectively coordinate tissue-scale behaviors. Tissue-scale calcium waves are one such signal. Intercellular calcium waves appear during the synchronized movements during wound healing (1, 2) and tissue development (3,4,5,6). Alternatively, supracellular cues can be mechanical, where the tissue’s local mechanics can relay information to the cell about its environment. For example, anisotropic stresses caused by a tissue’s nematic structure (7) can position within a tissue important biological phenomena including apoptosis (8), limb budding (9,10,11), and accumulation of neural progenitor cell defects (12). Many of these phenomena that are co-local with defects in a tissue’s nematic order, including cell death, cell extrusion, and limb development, simultaneously exhibit calcium spikes (13,14,15). Yet, how calcium signaling and nematic alignment establish collective cues in epithelia has only been considered in isolation.

A calcium flux can be triggered in a cell from a mechanical stimulus (e.g., wounding) or a chemical signal (1,16). When a sufficiently large increase in intracellular Ca²⁺ is triggered, an intercellular Ca²⁺ wave can propagate. Depending on the biological context, a global wave moves through tissue in two ways: intercellularly through gap junctions or through released extracellular signals such as ATP (17).

Local cell-cell coupling and tissue-scale structure are inherently connected to cell shape and cell-cell alignment. At the tissue scale, the degree of cell-cell alignment can vary. Elongated cells can organize like liquid crystals, forming locally well-aligned “nematic” domains (7,18). The active nature of epithelia spontaneously creates disorder in the form of motile topological defects, which has been well captured through the framework of active nematohydrodynamics (7,8,19,20). Anisotropy is not only important for mechanics, but also for signal propagation. For example, it is well appreciated that highly specialized cells, such as myocytes, leverage their anisotropy to directionally propagate signals at the tissue scale and perform their functions (21,22). Yet, how nematic order couples to the propagation of calcium waves in epithelial tissues is unknown.

We show that the speed of calcium waves across an epithelial tissue depends on the angle between a cell’s local nematic alignment and the orientation of the wave. We found that elongated RPE-1 cells have increased connectivity laterally with their co-aligned neighbors, leading to calcium waves propagating fastest “perpendicular” to the tissue’s nematic director. Further, we found that chemically inhibiting gap junctions and increasing cell density, respectively, decrease and increase wave speed. We frame our experimental observations with a simple model that posits that the propagation of calcium signals from one cell to its neighbors is dominated by anisotropic cell-cell coupling. Our model is able to recapitulate our experimental results, as well as predict how calcium waves move in tissues with nematic defects. Waves emanating from point wounds on aligned tissue evolve into ellipsoidal contours, whereas point wounds on nematic defects inherit symmetries from the underlying nematic: trefoil for −1/2 defects and circular profiles with a translating center of mass for +1/2. Our results show how a tissue’s nematic alignment, not spatial distance from a wound alone, controls when a cell receives a calcium signal. Our results demonstrate the importance of considering cellular nematics when interpreting how information propagates across epithelial tissue. Our quantitative description, of how topological defects impact calcium wave movement, elucidates how diverse developing tissues that exhibit calcium waves during morphological changes and wound healing collectively communicate.

Materials and methods

Cell culture

RPE-1 cells were maintained in Dulbecco’s Modified Eagle Medium with 1000 mg/L glucose, L-glutamine, and pyruvate (Thermo Fisher, cat#: 11885092), supplemented with 10% (v/v) fetal bovine serum (Fisher Sci., cat#: MT35011CV), and 100 μg/mL penicillin-streptomycin (Sigma, cat#: P4333). Cultures were incubated at 37°C in a humidified atmosphere with 5% CO₂. All experiments were performed using cells at or below passage 30.

Substrate preparation

Fabricating nematically aligned tissues

Microgrooves were created on the base of surface-treated, sterile 6-well tissue culture plates (Sigma, cat#: CLS3516) by gently dragging 2000-grit sandpaper across the well surface using light thumb pressure. This led to grooves been formed between 50 and 150 μm apart. Cells were seeded onto the grooved substrates and allowed to grow until confluence over several days. As cells adhered and proliferated, they aligned along the direction of the grooves, forming a tissue with local long-axis alignment. To vary cell density, the culture duration was varied: shorter growth periods produced lower densities, whereas extended growth periods yielded higher densities. For imaging cells with confocal microscopy (Fig. 2, E and F; Fig. S5, A and C), cells were seeded into 35-mm uncoated glass-bottom dishes (Fisher Sci., cat#: NC1844456) and allowed to grow to confluence. For immunofluorescence staining, cells were seeded onto a coverslip and allowed to grow to confluence.

Figure 2.

Gap junctions have nematic alignment and facilitate anisotropic transport of calcium. (A) Gap junctions in an epithelial tissue display structural anisotropy. (B) Immunofluorescence image of a cell monolayer showing gap junctions (green) and nuclei (blue). Gap junctions are distributed along the cell boundary, and nuclei are aligned with the cell’s major axis. (C) Using an angular FFT, we find the gap junction’s structural orientation is nematic and closely follows the orientational order of the tissue determined from the elongated cell nuclei. Nineteen replicates containing a total of 3800 cells were analyzed. (D) The angular difference Δθ between the gap junction structural orientation and nuclear orientation is strongly peaked at 0, ±π, indicating strongly coupled nematic alignment between the gap junctions’ structure orientation and cell orientation. (E and F) Time series of wave propagation parallel and perpendicular to the nematic director. (G and H) Characteristic profiles of calcium intensity between neighboring cells for (E) and (F), respectively. (I) Calcium travels slower parallel to the nematic director than perpendicular, shown by a longer lag time between intercellular calcium spikes. (J) Inhibiting gap junctions slows and then abrogates calcium transport. Gap junctions are inhibited with meclofenamic acid (MA), and these experiments were performed with waves parallel to the nematic. Each condition consists of three experiments.

Preparing a 1D line of cells

To generate 1D lines of cells, cells were seeded at very low density onto the microgrooved substrates described above and incubated for 24–48 h. After this period, wells were screened to identify those in which cells had aligned and proliferated along a single microgroove, forming a continuous line of cells. Approximately 10% of wells exhibited this pattern and were selected for further experiments. Wounds were scratched perpendicularly to the line at one edge.

Preparing a gap within a tissue

To create a gap within cell tissues, a silk sewing pin (Amazon) was used to create a wound across the aligned tissue and then washed to remove dead cells. The plates were then labeled with Calbryte-520AM. After 1 h the tissues with the gap were scratched within 500 μm of the gap to induce a wave. The wave from the initial wounding goes away within first 10–20 min after scratching and is not present when second wave near the gap is induced.

Reagents

Intracellular calcium labeling

Intracellular calcium was visualized using Calbryte-520AM (Fisher Sci., cat#: NC1812243). The dye was initially dissolved in DMSO and subsequently diluted to a final concentration of 5 μM in Hanks’ Balanced Salt Solution supplemented with calcium and magnesium (HBSS + Ca; Fisher Sci., cat#: J67763.). Probenecid (Sigma, cat#: P8761) was also added to this solution for a final concentration of 1 mM. Cells were incubated with 1 mL of the 5 μM Calbryte-520AM solution per well for 1 h at 37°C. After incubation, cells were washed with phosphate-buffered saline (PBS) and then maintained in HBSS + Ca. Imaging was performed immediately after dyeing, with HBSS + Ca used as the imaging medium unless otherwise specified.

Gap junction inhibition

Meclofenamic acid (MA; Thermo Fisher, cat#: J60484) was first dissolved in sterile deionized (DI) water to prepare a 1 mM stock solution. This stock was further diluted in HHBS + Ca to yield working concentrations of 5 μM, 15 μM, 25 μM, 35 μM, and 50 μM, as required for the experiments. Dishes were first incubated with Calbryte-520AM for 1 h and then washed before being incubated with MA for 20 min before imaging without washing.

Immunofluorescent staining

Coverslips containing confluent tissue cultures were fixed with 4% paraformaldehyde (Fisher Sci., cat# AAJ19943K2) for 15 min. Between each step, coverslips were washed 3× in PBS. After fixation, cells were permeabilized using 0.1% Triton X-100 (Fisher Sci., cat#: PI28314) for 10 min. Nonspecific binding was blocked using 1% bovine serum albumin (Fisher Sci., cat#: AAJ61089AP) in PBS for 30 min.

Samples were then incubated with a 1:1000 dilution of Anti-Connexin 43 Antibody produced in rabbit (Sigma, cat#: C6219) overnight at 4°C in the dark, using a box lined with wet paper towels. The following day, a 1:500 dilution of Anti-Rabbit IgG (H + L), CF568 antibody produced in chicken (Sigma, cat#: SAB4600426) and a 1:400 dilution of Phalloidin (Thermo Fisher, cat#: A12379) were applied for 1 h in the dark. Coverslips were mounted onto glass microscope slides using ProLong Diamond Antifade Mountant with DAPI (Thermo Fisher, cat#: P36966) and left to cure in the dark for 24 h at room temperature.

Tissue wounding

The cells were wounded during image acquisition on the microscope. A silk sewing pin (Amazon) was glided along the cell surface to create linear wounds. Point wounds were made by lightly touching the cell surface with the ball end of the pin.

Microscopy and imaging

For all experiments, wounds were created during image acquisition, and acquisition continued for at least several minutes up to 10 min.

Fluorescence and phase contrast imaging was performed using a Nikon Ti2 inverted microscope equipped with Nikon Elements software for image acquisition and an OKO CO₂ environmental enclosure to maintain physiological conditions. Fluorescence excitation for wounding experiments was achieved using a 480-nm excitation line via the GFP setting of the Sola Light Engine. Immunofluorescence excitation was achieved using a 480-nm excitation line via the GFP setting (actin), a 560-nm excitation line via the mCherry setting (Connexin-43), and the DAPI setting (Nuclei). Images were acquired using a Nikon Plan Fluor 4×/0.13 NA objective (WD 17.2 mm), a Nikon Plan Fluor 10×/0.30 NA Ph1 DLL objective, or a Nikon Plan Fluor 20×/0.5 NA 0FN25 Ph1 DLL objective. Phase contrast imaging was performed using the 10× objective to visualize cell morphology. Immunofluorescent images were captured with the 10×/objective. Time-lapse imaging was conducted at a frame rate of 500 ms per frame. Images were captured using a Hamamatsu ORCA-Flash4.0 LT digital CMOS camera (model C11440).

Confocal imaging was performed using a Nikon AX-R system mounted on a Ti2-E inverted microscope (Nikon Instruments). Samples were maintained under environmental control during imaging at 37°C and 5% CO₂ using a stage-top incubator. Excitation was achieved using a 488-nm diode laser at 25% laser power. A Nikon 40×/1.15 NA water-immersion objective was used with deionized water as the immersion medium. The resonant scanner was used for image acquisition. The pixel size was 0.12 μm. The pinhole was set to 2.1 Airy units. Emission was collected over a spectral detection window of 499.0–551.0 nm using a GaAsP detector with a gain setting of 1.5. Time-lapse imaging was conducted at a frame rate of 250 ms per frame.

Laser ablation

Confocal images were acquired on a Dragonfly 620-SR (Andor Technology, Belfast, UK) attached to a Nikon Ti2 stand. Sample was imaged with a 20 ×0.5NA objective lens. Calbryte-520am was excited with a 488-nm solid state laser and emission light filtered through a single bandpass 521/38-nm emission filter and onto a ZL41 sCMOS detector (Andor Technology).

Calcium waves were generated by wounding individual cells with a galvo driven MicroPoint 4 (Andor Technology). The MicroPoint 4 utilized a N2 pulsed laser directed into a dye cell with 435-nm coumarin dye, in turn creating a focused high-energy beam projected into the sample through the episcopic light path. A 460-nm long-pass dichroic mirror in the Ti2 filter turret allowed for simultaneous cell wounding and calcium imaging. The 435-nm beam was pulsed into the sample at 18 Hz. Number of pulses varied from five to 10 within individual cells.

Image analysis

Orientation fields were analyzed using the OrientationJ plugin for Fiji, which calculates local orientation and coherence based on the structure tensor method (23, 24). A custom Python script was used to overlay the orientations onto the phase contrast images to display the distribution of the orientation angles.

To measure wave propagation speed in response to linear wounds, kymographs of signal intensity were generated in Fiji (Fig. S2 B). To generate kymographs, a rectangular region of interest (ROI) was drawn along the axis of interest using Fiji (Fig. S2 A). The “Reslice” tool was applied from top to bottom, generating a time-lapse stack of the intensity along the ROI. A Z-projection (average intensity) of the resliced stack was then computed to produce the final kymograph (Fig. S2 B). An intensity threshold was applied to the kymographs to enhance contrast between background and wavefront signal (Fig. S2 C). The leading edge of the kymograph was then fit to a straight line. The inverse of the slope of this line was taken as the wavefront speed (Fig. S2 D).

For point wounds, a separate custom Python script was used to track wavefront progression over time. Two selected frames were binarized (Fig. S2, F and G), and the earlier frame was subtracted from the later one to generate a ring representing the spatial extent of the wavefront displacement during that time interval. Small objects were removed to eliminate noise, and a light dilation operation was applied to fill small holes and smooth the wavefront (Fig. S2 H). Radial widths of the resulting ring were then measured as a function of angle. Wave speed at each radial angle was computed by dividing the radial width by the time interval between the two frames (Fig. S2 I).

Intensity profiles were measured using a custom Python script by drawing a horizontal line across the center of the cell of interest. Fluorescence intensity along this line was tracked over time to generate intensity-versus-time data. In some cases, the intensity values along the line were averaged at each time point to produce intensity-versus-time plots (Fig. 2, G and H). In other cases, the full intensity profile along the line was retained to generate distance-versus-intensity plots (Fig. S5, B and D).

Aspect ratios of individual cells were determined in Fiji by measuring the lengths of the long and short axes. Ten cells were randomly selected per image for these measurements. Cell density was quantified by drawing a small rectangular ROI on the image using Fiji. The number of cells within this ROI was manually counted and divided by the ROI area to obtain density values. To analyze local tissue organization in Fig. S1 B, cell centroids were manually marked in each field of view. These centroid coordinates were input into a custom Python script to generate Voronoi tessellations, from which neighbor relationships were extracted.

Flow speed

The tissue’s flow speed (Fig. S3) was measured using particle image velocometry in MATLAB. Using 50% overlap, we performed three passes of window sizes 128 × 128, 64 × 64, and 32 × 32.

Structural anisotropy of gap junctions and nuclei alignment

The structural anisotropy of the gap junctions was analyzed in Python by computing the 2D fast Fourier transform (FFT)of the image. The angular power spectrum was obtained by averaging FFT power in different angular bins. We applied a low-pass filter k_cutoff = 6 to remove high frequency noise. Each sampled wavevector θ_k was converted to a real space vector θ_gap = θ_k + π/2. The orientational alignment of cells in Fig. 2 was determined by fitting cell’s elongated nuclei to an ellipse and extracting the major-axis angle. Nuclei were detected using Otsu thresholding, and OpenCV was used to obtain the contour. The orientation PDFs for gap texture and nuclei are averaged across 19 different replicates, containing a total of 3800 cells. The nematic director in experiments can be at an arbitrary angle in the lab frame, so to average our results meaningfully across data sets oriented at different directions, we compute the angle ${\theta }_0=\frac{1}{2}\text{arg}⟨e^{2i\theta }⟩$, rotate both the gap junction and nuclei angles by −θ₀, and pool the rotated raw angles to then built the final master PDF of gap junction structure orientation and nuclei angles. For each experimental image, the relative misalignment Δθ = θ_nuc − θ_gap was obtained, and the PDF was plotted. The relative misalignment PDFs from 19 different replicates are summed together to create one master PDF.

Statistical analysis

Statistical analyses were performed using Python. In Fig. 1 I, a Mann-Whitney U test was used to determine which wave speeds were statistically significant from those traveling parallel to the nematic director (θ = 0). Statistical significance was considered at p < 0.01. In Fig. 1 L, a paired t-test was used to compare parallel and perpendicular speeds of point wounds. Statistical significance was considered at p < 0.001. All experiments were performed with at least three replicates, measuring wave fronts propagating over a minimum distance of 250 μm from the wound. Where applicable, error bars represent the standard deviation.

Figure 1.

Nematic director controls speed of calcium waves initiated by linear and point wounds. (A and B) Phase contrast images of RPE-1 cells grown on a plastic culture dish (A) and a microgrooved substrate (B), overlaid with orientation fields. (Inset) RPE-1 cells are aligned with their neighbors. (C) Linear wounds were created by dragging a pin across the tissue, and localized point wounds were produced by gently tapping the surface with a rounded tool. (D) The nematic director $\hat{n}$ is the coarse-grained orientation field of the cell’s long axis. We define θ as the angle between wavevector $\hat{k}$ and nematic director $\hat{n}$ (from 0 to π/2). (E) Schematic and (F) time series of a linear wound that initiated a wave traveling parallel to the nematic director (θ = 0). (G) Schematic and (H) time series of a linear wound that initiated a wave traveling perpendicular to the nematic director (θ = π/2). (I) Experimental measurements of wave speed (gray disks) as a function of orientation θ. Error bars represent the standard deviation of the binned wave speeds. Waves travel significantly faster perpendicular (θ = π/2) than parallel (θ = 0) to the nematic director. Red line represents the model. Cell diagrams demonstrate the wave propagation direction with an arrow, relative to the tissue organization. Stars represent wave speeds statistically different from the θ = 0 bin using a Mann-Whitney U test, where ^∗∗ and ^∗∗∗ indicate p < 0.01 and p < 0.001, respectively. (J) Schematic and (K) time series showing outward, wave propagation from a point wound is anisotropic and depends on the nematic orientation. The polar angle ϕ is defined counterclockwise from the nematic director in the vertical direction. The angles ϕ = 0, π are parallel to the director; ϕ = π/2, 3π/2 are perpendicular to the director. (L) Quantification of wave speed from a localized wound in the ϕ = 0, π/2 directions. Waves propagate faster in the π/2 direction. ^∗∗∗ indicates p < 0.001 using a paired t-test. Wave speeds have been averaged radially. (M) Measured radial wave speed (disks) and rolling average (line, window size = π/9) from a point wound. The wave travels fastest perpendicular to the nematic axis (π/2, 3π/2).

Simulation details

We simulate Equation 1 using the finite element solver COMSOL. We arbitrarily set r = 1 and integrate the equations on a 400 × 400-sized square domain for 30 time units. We set D_‖ = 0.5 and D_⊥ = 2, which corresponds to a velocity v_⊥/v_‖ = 2 and mirrors the experiment (Fig. 1 I). When simulating the dynamics of point-wound and scratch-wound-initiated waves interacting with defects, we avoid singularities in $\hat{\mathbf{n}}$ by smoothly reducing the anisotropic contributions of the diffusion tensor to zero via a Gaussian divot with unit standard deviation located at the defect core. This region is small compared with the domain size. Thus, the large-scale dynamics we display are largely unaffected by this regularization.

Results

Purely nematic tissues form on microgrooved substrates

Retinal pigment epithelial cells (RPE-1) are an ideal cell model to investigate the interplay between nematic cell alignment and calcium wave propagation in an epithelial tissue. RPE-1 cells have a rod-like morphology making their orientation easily apparent. RPE-1 cells have a tendency to locally orient with their neighbors. Cell-cell interactions lead to neighborhoods of aligned cells, but confluent unconfined RPE-1 tissue remains globally disordered (Fig. 1 A) (7,25). We engineered long-range nematic order in the tissue by seeding cells on sparsely microgrooved substrates formed by lightly scratching a plastic-bottom dish with sandpaper. Scratches formed between 50 and 150 μm apart (Fig. S1 A). Cells on these substrates all oriented their long axis with the direction of the microgrooves (Fig. 1 B), suppressing the nematic defects seen on flat smooth substrates. Aligned cells still retain a characteristic epithelial packing pattern with an average of six neighbors (Fig. S1 B). With purely aligned tissues, we could address whether the relative angle between the cell’s nematic director $\hat{\mathbf{n}}$ and the direction of wave propagation impacts the wave speed (Fig. 1 I).

Speed of scratch-induced calcium waves depend on a tissue’s nematic director

The aligned tissues were incubated with Calbryte-520AM, a cell-permeable fluorescent indicator of intracellular calcium levels. We scratched a line across the plate of cells with a sewing pin to create linear wounds (Video S1) and measured the changes in fluorescence in the cells over ∼1 min. At this short timescale the tissue is a nearly static nematic layer: cell speeds in our tissues were measured to be 0.2 μm/h (Fig. S3). Linear wound experiments were performed at different angles relative to the global nematic orientation $\hat{\mathbf{n}}$ (Fig. 1 D). The mechanical wounding of the tissue resulted in a calcium flux in cells next to the wound and subsequent outward propagation of a calcium wave front. For linear wounds, a tissue-scale linear wave moved out perpendicular to the scratch (Fig. 1, E and G). We identified that the speed of the calcium wavefront depended on the cell’s orientation with respect to the incoming wave (Video S2). The wave speed is maximal when the wound is aligned along the nematic director θ = π/2, i.e., scratch parallel to the cell’s long axis, and minimal when the wound is perpendicular to the director θ = 0, scratch along the cell’s short axis, with intermediate speeds for oblique orientations (0 < θ < π/2) (Fig. 1 I). To control for a possible impact of the grooves themselves on wave speeds, we wounded unaligned tissues grown on flat nonscratched surfaces. Waves moved with comparable speeds on purely flat substrates (Fig. S4 A).

We next asked if the speed of a radial wavefront exhibited the same dependence on cell nematic orientation as a linearly propagating wavefront. We locally wounded a uniformly aligned nematic tissues with a small spherical tip to create point wounds (Fig. 1, C and J, Video S3). Similarly to linear wounds, point wounds triggered cells adjacent to the wound to light up and subsequently a wave propagated away from the wound. Waves traveled slower in all directions compared with the linear wound. Still, the calcium wave progressed faster perpendicular to the nematic compared with parallel to the nematic (Fig. 1, L and M).

Anisotropic coupling exists between neighbors

Next, we investigated the origin of the anisotropy in the speed of a calcium wave across an aligned tissue. Gap junctions have a well-established role in propagating tissue-scale calcium waves (17). We hypothesize that wave speed anisotropy arises from greater connectivity laterally between elongated cells. Schematically, we illustrate the logic behind our hypothesis in Fig. 2 B, showing how increased contact area naturally leads to stronger coupling through a greater number of gap junctions. We directly explored cell-cell connectivity with immunofluorescence staining of Connexin-43, a gap junction protein (Fig. 2, A and B). We found that gap junctions are organized into stripe-like patterns oriented parallel to the tissue nematic director (Fig. 2, C and D). Gap junction orientation was quantified using an angular Fourier power spectrum analysis, and the cell orientation was independently determined from the alignment of elongated nuclei (Fig. 2 C), which has been shown to correlate with the cellular major axis (26,27). Because the Fourier power spectrum reflects spatial intensity variations, its angular maxima occur perpendicular to the underlying stripe orientation; accordingly, we rotate the extracted angular distribution by π/2 to enable direct comparison with nuclear alignment. This characterization supports our hypothesis that the organization of gap junctions in relation to tissue structure preferentially confers strong coupling between minor-axis neighbors.

We further quantified anisotropy at the cell scale by examining the calcium dynamics of pairs of cells that are either connected end to end (Fig. 2 E , Video S4) or connected side to side (Fig. 2 F , Video S5). We observe that in both cases, the interior of a given cell activates nearly uniformly. That is, upon stimulation, internal wave propagation is rapid—much faster than the wave speed at the tissue scale; we further quantify this in the supporting material (Fig. S5). Strikingly, however, the time between spikes for the two pairs is different. End-to-end connected cells that propagate a wave parallel to the nematic director show a large time lag between spikes compared with that of the lateral pair that propagate a wave perpendicular to the nematic director. Fig. 2, G and H plots the cells’ intensity at their centers over time, demonstrating this difference. From this, we conclude that gap junctions control the time lag of calcium spikes between cells, thereby determining the wave speed at the tissue scale.

Video S4. Time lapse of calcium wave propagating across two lateral neighbors

Cell outlined in green increases fluoresence first, and cell outlined in blue increases in fluoresence second.

Video S5. Time lapse of calcium wave propagating across two tip-to-tip neighbors

Cell outlined in green increases in fluoresence first, and cell outlined in blue increases in fluoresence second.

Gap junction coupling is needed for signal propagation

To test the importance of gap-junction-mediated coupling at the tissue scale, we experimentally decreased gap-junction activity and measured wave speed. We titrated the inhibition of gap junctions using MA, which is reported to block gap junction communication in RPE cells (28). A linear wound was performed at an angle of 0 radians. At concentrations of 5 μM and 15 μM the wave can still propagate, but it moved slower than in the control where no inhibitor is added (Fig. 2 J). At and above 25 μM, we did not observe any tissue-scale waves, and only the cells that contacted the wound lit up (Fig. S6).

Tissue density positively correlates with calcium wave speed

Upon measuring that decreased gap junction activity slows the wave, we asked if conversely increasing cell-cell connectivity would trigger increased wave speeds. We increased the amount of cell-cell connectivity by modulating tissue density (Fig. 3 F). From a purely geometrical argument, increasing the density of packed particles increases the average number of contact points between neighbors. Increasing the anisotropy of packed particles preferentially increases contact points laterally between neighbors. We measured that as tissues get denser, cells width decreased (Fig. 3 B) and cell anisotropy increased (Fig. 3 C), highlighting that cells were getting crowded together laterally (Fig. 3 E).

Figure 3.

Cell density promotes connections, increases wave speed. (A and B) As cell density increased, the cell length remains constant, but the cell width decreased. (C) Hence, aspect ratio of cells increased with increasing density. (D) Calcium wave speed increased with cell density. (E) Schematic representing that increasing density in tissues increases lateral contacts. Gray squares represent areas with no cells. Lower density tissues are not confluent. (F) RPE-1 cells at three different densities. As the cells become denser, there is less free space between cells and more cell-cell contact. Schematic (G) and times series (H) of wave propagation across a 1D line of cells. Schematic (I) and time series (J) of wave propagation across a gap of no cells. All experiments in this figure are performed such that the calcium wave traveled parallel to the nematic director.

We performed linear wounds oriented at θ = 0 radians in tissues of varying density. The calcium wave moved faster in denser tissues (Fig. 3 D). This revealed that a global wave moving along the nematic director is sped up by increased lateral connections along a cell’s short axis.

To test the limits of wave propagation through cell-cell contact, we performed linear wounds on single-file lines of RPE-1 cells (Fig. 3 G) that formed along a single groove in low-density plates. In this condition, only a minimal contact existed between cells at their tips, compared with our experiments in dense tissue that had lateral and tip contacts with neighbors. The wave moved slower in the line configuration than in the 2D tissues (Fig. 3 D), but it was still able to propagate.

Finally, to further confirm the impact of cell-cell coupling on wave speed, we performed a linear wound experiment on a tissue that had a gap between two populations of aligned cells (Fig. S7). Consistent with published results (1), we saw that a wave triggered in one population could stimulate a calcium wave in the second population (Fig. 3 I, Video S6). However, the time it took for the unconnected population of cells to respond and light up was slower than the time for the signal to propagate in contacting cells (Fig. 3 D). This demonstrated that although gap junctions mediate a fast signal between cells, a diffusible signal can also stimulate a wave moving over longer timescales. We did not observe an extracellular diffusion-driven wave when inhibiting gap junctions, suggesting a need for some cell-cell junction activity to respond to the extracellular signal.

Video S6. Time lapse of calcium wave propagating across a (A) 1D line of cells and (B) across a gap of no cells

The wave travels faster along the 1D line than across the gap. The line video stops after 44 s, whereas the gap continues through 91 s.

A calcium wave moving through a nematic tissue can be modeled by an anisotropic diffusion in a reaction-diffusion PDE

We turned to modeling to develop a more quantitative understanding of how calcium waves move in our structured tissue. We modify a well-studied model of nonlinear front propagation, the KPP equation (29).

$$\frac{\partial u}{\partial t}=\nabla ·\left(\mathbf{D}\nabla u\right)+ru(1-u),$$ (1)

where D is the diffusion tensor, r is the reaction rate, and u is a scalar field, which we take to represent the calcium concentration. Upon perturbation to a uniform state u = 0, the system rapidly transitions to a stable value u = 1. Spatially localized perturbations propagate outward as a well-defined solution with a characteristic velocity $v\sim \sqrt{Dr}$ (30). In this simplified framing of the experimental dynamics, intercellular coupling mediated by gap junctions and extracellular communication through the media are collectively expressed by the anisotropic diffusional transport of u, whereas dynamics leading to the rapid spike in intracellular calcium are approximated by the nonlinear reaction ru(1 − u). We express the diffusion tensor as follows:

$$\mathbf{D}=D_{\Vert }\hat{\mathbf{n}}\otimes \hat{\mathbf{n}}+D_{\perp }(\mathbf{I}-\hat{\mathbf{n}}\otimes \hat{\mathbf{n}}),$$ (2)

where D_‖ and D_⊥ are the diffusivities parallel and perpendicular to the nematic director $\hat{\mathbf{n}}$, respectively, and ⊗ represent outer products of the units vector. The nematic director $\hat{\mathbf{n}}$ is a headless vector encoding the orientation field of the tissue. Parallel and perpendicular diffusivities have successfully described anomalous Brownian motion in nematic liquid crystals (31,32,33,34); however, extending a nematic-like diffusion tensor to a reaction-diffusion system is unexplored in the context of epithelia to our knowledge.

Before presenting predictions of wave propagation by numerically integrating Eq. 1, we first estimate the impact of anisotropy on wave speed analytically. In the sharp interface limit, neglecting the chemical wavefront’s curvature, and assuming uniformity in the underlying nematic, the wavefront velocity is given by

$$\begin{array}{l}\hfill v=2\sqrt{r{D}_{\text{eff}}},\\ \hfill D_{\text{eff}}=D_{\Vert }{\cos }^2\theta +D_{\perp }{\sin }^2\theta ,\end{array}$$ (3)

where the effective diffusivity depends on the angle $\theta =\arccos (\hat{\mathbf{k}}·\hat{\mathbf{n}})$ between the wavevector $\hat{\mathbf{k}}=\nabla u/|\nabla u|$ and the nematic director $\hat{\mathbf{n}}$. We show the full derivation of Eq. 3 in Eqs. S1–S10. We fit Eq. 3 to the experimental data in Fig. 1 I and find that it captures the leading-order effect: maximal velocity perpendicular to the nematic director and minimal velocity parallel to the nematic director. Using two free fit parameters for rD_‖ and rD_⊥, we obtain the ratio of the diffusivities D_⊥/D_‖ ≈ 4.2. Because our fit finds that the perpendicular diffusivity is roughly four times the parallel diffusivity, the perpendicular wave speed v(θ = π/2) should be roughly twice the parallel wave speed v(θ = 0), which is exactly what we observe in the experiment (Fig. 1 I). We attribute deviations from this idealized theory in Fig. 1 I to the complex microstructure of the tissue, which we discuss further in the conclusion.

Wave front encodes topological information

Our purely nematic tissues seeded on grooved substrates allowed us to identify the relationship between a cell’s orientation with respect to a calcium wave and the global wave speed. However, nematic defects exist in the tissue where regions of cells are locally aligned but globally disordered. In unconfined RPE-1 tissue, nematic defects were separated by 500–1000 μm, with the distance between defects increasing with increased density, consistent with published studies (25). Important biological processes are known to localize at nematic defects, such as apoptosis (8) and budding (20,35). These events can also trigger calcium signaling (13,14,15).

We therefore leveraged our model to simulate the evolution of a calcium wavefront from a wound near a topological defect. We integrate Eqs. 1 and  2 subject to circular and linear initial conditions to emulate experimental linear and point wounds. Simulations of a linear wound in a uniformly aligned nematic (Fig. 4 A, Video S7) recover similar linear translation of the chemical wavefront as was observed in the experiments (Fig. 1, E and G). Similarly, a point wound in a uniform nematic (Fig. 4 D) assumes an elliptical-like wavefront shape, as was found in the experiment shown in Fig. 1 J. However, introducing topological defects dramatically changes the behavior of chemical waves. An initially flat plane wave traveling through a topological defect does not remain flat. As the wave travels, the nematic director around it changes—hence changing the angle between the wave vector and nematic—altering the local wave speed and distorting the wavefront (Fig. 4, B and C).

Figure 4.

Nematic texture gives shape to chemical wavefronts. We simulate the evolution of a chemical wave traveling through different nematic textures with the experimentally measured ratio of perpendicular to parallel diffusivities D_⊥/D_‖ ≈ 4. (A) A plane wave maintains its shape and travels at a constant speed through a uniformly aligned nematic. (B and C) However, a plane wave traveling through a (B) −1/2 and (C) +1/2 topological defect has its wavefront distorted. The angle between the wavevector and nematic director changes while the wave travels, modulating the local wave speed. (D) A point source emitted outward in a uniformly aligned nematic assumes an elliptical wavefront shape. (E and F) Wavefronts from point sources in topological defects encode the symmetry of the defect. (E) A wavefront emitted outward from a −1/2 defect has trefoil symmetry. (F) And, a wavefront emitted outward from a +1/2 defect assumes a circular shape and translates to the right. (G) For the point source emitted outward in a +1/2 defect, the x-coordinate center of mass increases linearly, whereas the point sources emitted from a −1/2 defect and a uniformly aligned nematic do not translate. The y-coordinate center of masses does not translate. (H) The area enclosed by the wavefront increases faster from a point source emitted in a −1/2 defect and uniformly aligned nematic than from a point source emitted from a +1/2 defect.

We observed a symmetry-preserving relationship between nematic defects and the chemical wave front’s shape when initiated from a point wound coinciding with the defects’ cores (Video S8). We find that a wavefront emanating from a −1/2 defect exhibits a trefoil shape following the threefold symmetry of the underlying nematic (Fig. 4 E). And moreover, a wavefront emanating from a +1/2 defect displays a nearly circular shape with the centroid acquiring a steady shift of the contour’s center of mass to the right as the region emanates outward (Fig. 4, F and G). The broken polar symmetry of +1/2 defects is well studied in active nematohydrodynamics, but their impact on front propagation has not been documented to our knowledge. We further quantify the areas of the growing fronts and find the area enclosed by the wavefront grows at different rates depending on the tissue’s underlying nematic orientation (Fig. 4 H). The contour area grows slowest for a point source emitted from a +1/2 defect compared with a point source emitted at a −1/2 defect or in an aligned tissue.

Discussion

Biology self-organizes by closely coupling form, mechanics, and biochemistry. Deepening our understanding of how these physical processes interrelate across scales and cell types is an ongoing effort that will impact developmental biology and medicine. Here, we experimentally demonstrate that an interplay exists between the speed of calcium wave propagation and the tissue’s microstructure, specifically its nematic order. We conclude that the origin of wave speed anisotropy is the local coupling environment of cells, wherein elongated cells in a tissue more strongly couple to their lateral neighbors due to increased contact area. We developed a simple mathematical model that recapitulates our observations of how wave speed depends on this structure. Although our study focuses on the propagation of calcium waves through epithelial media, the underlying physical mechanism is generic and likely to apply broadly to other cell types and communication pathways. The impact of anisotropy could also be significant even in the absence of coherent, large-scale signaling waves, as we’ve studied here.

Living nematic epithelial tissues exhibit birefringence

The analogy between tissues and nematic liquid crystals has provided insight into collective dynamics of cells. Liquid crystals are birefringent, whereby the material has two different refractive indices for polarized light either parallel or perpendicular to the material’s internal molecular orientation. We demonstrate birefringence in epithelial tissues with a calcium wave moving at different speeds depending on its orientation relative to a tissue’s nematic director. Two differences, however, to consider between our system and nonliving materials are, first, that in our tissues, part of the propagation occurs via soluble agents, as demonstrated in Fig. 3, I and J, albeit at slower rates than gap junction coupling. Additionally, in liquid crystals, beam propagation occurs in 3D, whereas our system is being modeled in 2D. Still, our results strengthen the analogy of tissues as nematic liquid crystals.

Nematic defects lead to heterogeneity in tissue’s spatial response to wounding

We demonstrate numerically that nematic defects desynchronize a propagating calcium wave. The area inside the calcium wave contour grew at different rates, depending on the underlying tissue’s nematic orientation. Therefore, cells at the same spatial distance from the point wound received information at different times depending on if they were triggered on aligned tissues, +1/2 or −1/2 defects. The same mechanical wounding leads to different tissue-scale outcomes. Similarly, in propagating plane waves, the tissue structure distorts the wavefront, and cells equidistant from the wave’s initial starting point experience a wave passing at different times depending on their position relative to a nematic defect. This asynchrony in when neighboring cells receive a calcium signal could have functional importance in vivo. A calcium signal can trigger cells to constrict (36), and temporal variation in cell constrictions is important in guiding diverse shape changes during development (37).

Although we experimentally show that the calcium wave speed is impacted by global tissue structure, a limitation is that we lack an experimental validation of our model in the vicinity of defects. We experimentally explored the predictions of our model with laser ablation, which allowed us to precisely target the core of the defects. However the waves resulting from the limited damage done by laser ablation experiments resulted in slow waves, consistent with the point wound on an aligned tissue (Fig. 1, K–M). At short time and length scales, our results were inconclusive and are illustrated in Fig. S8. The clear trend we identified in Fig. 1 I, however, motivates the need for future work to improve experimental wounding. The model gives a quantitative prediction of dynamics around defects if the wave speed and anisotropy present experimentally in large scratch wounds (Fig. 1 I) can be replicated experimentally for localized circular wounding.

Beyond nematic order

By introducing nematic order-dependent diffusivity, we predict the dependence of wave speed on the relative angle between the wave front and director. Although our model qualitatively shows that maximal and minimal wave speeds, respectively, correspond to waves moving perpendicular and parallel to the director, Fig. 1 I shows quantitative differences between theory, Eq. 3, and observations at intermediate angles. We posit that this discrepancy is due to the complexity of the tissue’s microstructure. For example, although the cells are elongated and aligned with their neighbors, the disordered packing of the tissue exhibits a highly hexatic structure in addition to nematic order (Fig. S1 B). Recent studies of epithelial flows have quantified cellular shape and packing order parameters, incorporating them into active hydrodynamic theories (38,39,40). Our work suggests that this structure influences cellular communication in purely aligned tissues and makes a theoretical prediction of a role of topological defects in bending wave fronts. We predict wave speed anisotropy might even serve as a measure of the underlying structure. Taken together, our observations suggest an exciting line of experimental and theoretical inquiry that further investigates the relationship between wave propagation and the various forms of order that exist in tissues.

Mechanistic models

In our experiments, we observed an overall decrease in the speed of point-wound-initiated waves compared with scratch wounds. This finding was consistent with the literature, which noted that the amount of initial wounding impacts wave response (41). We hypothesize that the decreased contrast in wave speed between parallel and perpendicular directions could be related to the diffusible signal that is released from the wound. Previous work has shown that calcium wave dynamics depend on the availability of extracellular ATP, which plays an excitatory role in the signaling dynamics. ATP is released endogenously by cells during calcium signaling but can also be introduced exogenously as an experimental control parameter (1, 42). In some cell types, the absence of a baseline level of extracellular ATP prevents the propagation of signals altogether (1). Cell injury releases freely diffusing ATP and contributes to the initial excitation. The geometry of the wound significantly influences this initial distribution. Comparing linear and point wounds, an initial circular distribution of ATP will diffuse radially and dilute more rapidly than a linear distribution. Thus, we posit that the slower wave speeds we observe are due to this extracellular effect. This is further corroborated by a wave speed dependence on initial wound size, with larger wounds.

Capturing these physical processes theoretically is beyond the capabilities of our simplified model, which evolves a single scalar value over time and omits biochemical details in favor of a focus on anisotropic diffusion. A future modeling direction is to explicitly capture the transport of specific molecular species, such as Ca²⁺, ATP, and IP₃ (43,44,45). In such a model, one could mimic experimental techniques by shaping initial extracellular ATP distributions.

Biology and topology

Examining the coupling between topology and chemical processes is an active area of focus, with researchers investigating both how chemical perturbations can precipitate structural rearrangements and how topology can shape reaction-diffusion processes (10,46,47). In particular, nematic defects localize important biological processes in various tissues, including cell extrusion in biofilms (8,20,35) and limb growth in multicellular life (10,46,48). In vivo, calcium waves are commonly observed emanating from such events (13). Beyond animal epithelia, plant tissue also exhibits chemical-topological feedback loops. For example, the growth hormone auxin is pumped in a directional manner along the axis of cells, which, in turn, grow to orient along those dynamic gradients (49,50,51). This feedback loop reinforces the gradients and spontaneously colocalizes auxin maxima with topological defects. Another example of chemical feedback in polar media is motile Dictyostelium cells, which use cyclic AMP signaling to collectively select a gathering site and cooperatively form multicellular structures (52). Taken together, these examples highlight how structural-chemical feedback mechanisms organize biological pattern formation. Our work suggests that anisotropic diffusion could be another tool used by biological systems to shape collective organization.

Conclusion

Our work uncovers an interplay between cell nematic orientation and calcium wave speed. We show how these two tissue-scale cues interact to impact long-range communication in epithelial tissues. Our work therefore provides an entirely new perspective on how physical and biochemical cues can be integrated to coordinate collective cell behaviors. This mechanism has relevance for the diverse range of tissues that display calcium waves during development and disease.

Data and code availability

Data are available from the lead contact upon reasonable request.

Acknowledgments

Research reported in this publication was supported by the National Institute of General Medical Sciences of the National Institutes of Health under award number R00GM136915. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. We thank Dr. Piali Sengupta for the RPE-1 cells. The development of the computational model was supported by Department of Energy BES no. DE-SC0022280 (M.M.N.). We also acknowledge the use of the optical and biomaterial facilities supported by NSF MRSEC grant no. DMR-2011846, the Brandeis Light Microscopy Core Facility RRID: SCR_025892 for equipment, and Dr. Andrew Stone for assistance.

Author contributions

H.G.Y. and A.C.W. designed research; A.C.W., B.C.S., and M.M.N. performed research; all authors analyzed data; all authors wrote the paper.

Declaration of interests

The authors declare no competing interests.

Editor: Timo Betz