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. Author manuscript; available in PMC: 2025 Nov 1.
Published in final edited form as: J Biomech. 2024 Sep 26;176:112342. doi: 10.1016/j.jbiomech.2024.112342

Energetic Scaling Behavior of Patterned Epithelium

Frank D Peters 1,2, Tasnif Rahman 1,2, Haokang Zhang 1,2, Leo Q Wan 1,2,3,4,*
PMCID: PMC11560681  NIHMSID: NIHMS2026689  PMID: 39342903

Abstract

Cellular monolayers display various degrees of coordinated motion ranging from the small scale of just a few cells to large multi-cellular scales. This collective migration carries important physical cues for creating proper tissue morphology. Previous studies have demonstrated that the energetics of the epithelial monolayer show a linear variation with time in conjunction with an arrest in monolayer motion after confluency. However, little is known about how the energetics of monolayer development are affected by confined geometries. Here, we demonstrate that micropatterned epithelial monolayers display a non-linear change in energetic variables, which coincides with the large-scale coordination of migration. This non-linear scaling behavior was further seen to be associated with the biased alignment of cells and cell-cell adhesion. These findings provide a new understanding of how developing epithelia may be impacted by different conditions in vivo.

1. INTRODUCTION

Cellular motion sees a wide variety of coordinated activities at different time and length scales. The collective migration of epithelial cells, in particular, plays a critical role in the development of tissue morphology such as invagination, looping, extension, and rotation (Ray et al., 2018; Weijer, 2009; Xi et al., 2017; Yip et al., 2018). Large-scale motion of groups of cells have been observed in many biological and pathological processes, such as zebrafish embryo gastrulation and tumor progression (Behrndt et al., 2012; Friedl et al., 2012). In developmental tissues, cells migrate and create tissues of specific geometries, such as tubular structures (Sanchez-Corrales et al., 2018; Xi et al., 2017), and these geometries, in turn, constrain the movement of the cells. In this regard, the latter process, i.e., how geometric constraints regulate multicellular morphogenesis, is still poorly understood.

On two-dimensional patterned and un-patterned substrates, epithelial cells tend to adhere and move in clusters, known as collective cell migration (Petitjean et al., 2010; Tambe et al., 2011; Zehnder et al., 2015). Modes of migration can range from small clusters of cells within large monolayers to large-scale motion of the entire monolayer itself (Lin et al., 2018). With the continuing increase of cell density, the cellular motion within the monolayer is generally seen to subside (Lin et al., 2021; Petitjean et al., 2010). This change in activity can be effectively described by proportional decreases of energetic parameters (Blanch-Mercader et al., 2018; Dunkel et al., 2013; Lin et al., 2021). From the velocity field estimated from the particle image velocimetry (PIV), parameters such as vorticity, kinetic energy, and enstrophy can be obtained. Kinetic energy is characterized by the velocity squared, describing the energy associated with cell translocation. Vorticity describes the degree and direction of swirling of the velocity field. Figure 1 demonstrates that vortexes of different handedness form with velocity vectors swirling in opposite directions and the negative and positive vorticity value stand for clockwise (CW, red) or counterclockwise (CCW, blue) motion, respectively. Enstrophy, in turn, is calculated from vorticity squared, reflecting the overall swirling behavior of cells, regardless of direction, as well as the general dissipation of kinetic energy. The decrease of these energetic parameters of the monolayers has been observed consistently with different cell types and even in bacterial cultures (Dunkel et al., 2013; Lin et al., 2021). Within epithelia, the natural formation of vortical structures swirling directionally (either in the CW or CCW direction) has been seen to decrease with time (Blanch-Mercader et al., 2018). The energetic trends of migration, however, have been primarily examined for the monolayers without geometric confinement. Cells constrained in geometrically defined environments would show different migratory behaviors, especially considering the intrinsic bias of the cells. This study, therefore, will examine the energetic behavior of patterned monolayers.

Figure 1: Velocity fields and vorticity.

Figure 1:

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Cells swirling in the counterclockwise direction (left; see the velocity v(x, y) indicated by black arrows) generate a positive (red) vorticity (ω). Inversely, the clockwise swirling of cells (right) generates a negative vorticity (blue).

When contacting geometric boundaries, cells exhibit specific behaviors that are not observed otherwise. It is well known that cells tend to align along geometric boundaries, known as contact guidance. Recent research shows that the migration of cells at boundaries is directionally biased. This is characterized as cell chirality, the handedness or left-right symmetry of the cells (Wan et al., 2011). Several cell types on micropatterns migrate in opposite directions on the appositional boundaries (Duclos et al., 2018; Wan et al., 2011; Yashunsky et al., 2022). Patterned monolayers of epithelial Madin Darby Canine Kidney (MDCK) cells have shown directed migration on the substrate boundary (Rahman et al., 2023; Worley et al., 2015) and later, at a higher cell density, developed into a coordinated motion of the entire monolayer (Doxzen et al., 2013; Luo et al., 2023; Rahman et al., 2023). Simulations have shown that the chiral torque of the cells within a confined epithelium further aids in generating a coordinated migration (Rahman et al., 2023; Yamamoto et al., 2020). This increase of motion with cell density contrasts the understanding of monolayer arrest from the energetic perspective.

Here, we aim to examine how collective migration affects the energetic scaling of patterned epithelial monolayers. Using PIV, we observe that MDCK ring patterned monolayers display a non-linear increase in kinetic energy in conjunction with the coordination of migration on the boundaries. This behavior has further been associated with the chiral alignment of cells and cell-cell adhesion. Our findings highlight a unique phenomenon within epithelial monolayers that could aid in understanding the development of tissues.

2. METHODS

2.1. Cell Culture

MDCK and C2C12 cells were cultured in Dulbecco’s Modified Eagle Medium (DMEM; Gibco) supplemented with 1mM Sodium Pyruvate (Thermo Fisher Scientific), 1% Penicillin/Streptomycin (Sigma-Aldrich), and 10% Fetal Bovine Serum (Seradigm) at 37°C with 5% CO2. Cells were trypsinized and seeded onto fibronectin-coated surfaces or 6-well cell culture plates (Cell Treat).

2.2. Microcontact Printing

The microcontact patterning is performed using gold-coated glass slides, as described previously (Zhang et al., 2022). A transparency photomask (CAD/Art Services Inc.) was used to transfer geometric features to a silicon wafer master mold using SU-8 2050 (Micro-Chem) photoresist using UV photolithography (OAI Contact Aligner). Polydimethylsiloxane (PDMS; Corning) stamps were created using this master mold. The stamps were then used to transfer the adhesive self-assembly monolayer (SAM), octadecanethiol (Sigma), to the gold-coated slides. The background was subsequently blocked through immersion of the slides in a non-adhesive ethylene glycol-terminated SAM (HS-(CH2)11-EG3, Prochimia). The patterned gold-coated slide was then washed with ethanol, dried, and treated with 50 μg/mL of fibronectin (Sigma) before cell seeding.

2.3. Cell Patterning and Chirality Analysis

Cells were seeded onto substrates at a low density (~100,000 cells/mL). Once ~50% confluency on rings was reached (30 minutes), excess cells were washed off with fresh media. When needed, ethylene glycol tetraacetic acid (EGTA; Sigma) was applied approximately 20 hours after seeding at a concentration of 0.5 mM or 1.0 mM for 16 hours. Using a custom MATLAB code (Zhang et al., 2022), phase contrast images were analyzed using a sub-region phase contrast intensity gradient approach to determine the angle of alignment (Wan et al., 2011). A singular frame of the video is determined to be chiral if the distribution of sub-region alignments is significantly different from the circumferential or tangential direction (see Fig. 4A–D), based on the circular t-test (Berens, 2009).

Figure 4: Chiral alignment of cells on the patterns correlates with non-linear energetic scaling.

Figure 4:

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(A) The local alignment of cells on a ring pattern displaying linear scaling behavior. (B) A rose plot showing the distribution of alignment is not significantly biased (ns; p > 0.05). (C) Local cellular alignment on a non-linear scaling ring pattern. (D) The distribution of alignment on this pattern was determined to be significantly chiral in the counterclockwise direction (*; p < 0.05). (E) The table breaks down the distribution of chirality of chirality within the patterns and energetic scaling behavior. By proportionality test, there is a significantly larger fraction of chiral rings in the non-linear scaling group, compared to that in the linear scaling rings (***; p < 0.001). (scale bars = 50μm)

2.4. Live Cell Imaging

Time-lapse imaging was initiated 24 hours after cell seeding onto micropatterns. The cells were maintained in the same media and culture conditions as mentioned above for the duration of the time lapse. Using the Keyence BZ-X microscope, live phase contrast images were taken every 10 minutes at 10x magnification for 12 and 24 hours, totaling 36 or 48 hours of culture time after cell seeding, respectively. The spatial resolution of 10x images corresponds to approximately 0.758 μm/pixel. Time-lapse videos were then output in individual video files per region of interest.

2.5. Particle Image Velocimetry (PIV)

The PIVLab (Thielicke and Sonntag, 2021) MATLAB application was used to generate velocity vectors from the time-lapse phase contrast images. For subregion motion vector interpolation two interrogation window sizes were used. The first pass was 64 x 64 pixels (~48 μm). The second pass was 32 x 32 pixels (~24 μm). Both passes utilized 50% overlap, leading to an overall distance of 16 pixels (~12 μm) between the final calculated vectors.

2.6. Energetics Analysis

All analysis beyond PIV was completed using custom-written MATLAB code. The vorticity was calculated for each PIV location within each frame. A finite difference approach was utilized to determine a local vorticity measure, as shown below. The vorticity of regions on the edge of the masked area was ignored.

ωij=12ΔX(vyi+1,j−vyi−1,j)−12ΔY(vxi,j+1−vxi,j−1) (1)

The vorticity was calculated over all elements of i and j in the x and y field of view, respectively. The first (second) term details the change in the y-component (x-component) of the velocity in the x-direction (y-direction). The terms are normalized by the grid spacings ΔX and ΔY of the vector space in their respective directions.

The Kinetic energy (E(r,t)=12|v(r,t)2|) was calculated from the velocity field (v), and enstrophy (Ω(r,t)=12|ω(r,t)2|) was calculated from the vorticity field (ω). The average value of each of these quantities was calculated for each frame. Energy dissipation was calculated by summation of the square of all symmetric parts of the velocity gradient tensor ∂vi/∂xj.

3. RESULTS

3.1. MDCK cell monolayers on ring micropatterns display non-linear trends in energetic scaling

Time-lapse videos of un-patterned monolayers of MDCK cells display a linear trend in energetic development (Fig. 2 A–D, Vid. S1), consistent with previous literature (Lin et al., 2021). The energetic scaling trend (Fig. 2 C) can be seen to decrease with time in a linear fashion (indicated by red arrows). The decreasing trend in kinetic energy (E) and enstrophy (Ω) highlights the natural progression of the monolayer to come to arrest with a significant decrease in cellular motion. To assess the change in monolayer motion, we can evaluate the length scale of swirling behavior from the slope of the energetic scaling graph (Fig. 2 C). The relative swirling size, E/Ω, aids to describe the overall translational to swirling behavior within the monolayer (Blanch-Mercader et al., 2018; Dunkel et al., 2013; Lin et al., 2021). For all un-patterned cases, a mean value of 24.3 μm (± 2.4 μm, n = 24) was found to persist over the entire time of culture (Fig. 2 D). Relative swirling size also acts to highlight the consistency of the overall slope in the E vs. Ω graph.

Figure 2: MDCK ring patterns display non-linear energetic scaling at later time points in monolayer development.

Figure 2:

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(A-B) A phase contract image of MDCK monolayer with no patterning and associated PIV analysis shows local velocity and vorticity. (C) Enstrophy and kinetic energy averages for every frame of the monolayer time lapse are plotted against each other (color indicates time; red arrows highlight the linear trend towards the origin point). (D) The relative swirling size obtained from (C) is plotted against time. (E-H) Linear energetic scaling of an MDCK monolayer seeded onto a ring patterned substrate. (I-L) Non-linear energetic scaling of an MDCK ring patterned monolayer. (K) The kinetic energy vs. enstrophy graph shows that at approximately 32 hours post seeding there is a change in overall scaling behavior (indicated by red arrow). (L) The relative swirling size of the monolayer additionally sees a significant change at later time points. (all scale bars: 50 μm, all arrows = 50 μm/hr).

Ring-patterned monolayers display two distinct trends of energetic scaling. Firstly, the patterned monolayers displayed a linear scaling regime (Fig. 2 E–H, Vid. S2). This behavior directly coincides with what is seen in the un-patterned monolayer with a persistent swirling size. Other ring patterns, however, display a non-linear biphasic energetic scaling behavior where E and Ω no longer decay simultaneously (Fig. 2 I–L, Vid. S3). The energetic scaling plot instead shows that towards the end of the 12-hour time-lapse imaging (i.e., 36 hours after cell seeding), the monolayer sees an increase in the overall kinetic energy (highlighted by the red arrow, Fig. 2 K). Additionally, the relative swirling size of the monolayer significantly increases at hour 36 (Fig. 2 L). Ring patterns were visually classified as non-linear based on the overall shape of the E vs. Ω graph. Of the 26 ring patterned monolayer videos, 65% (17) display linear scaling behavior, while 35% (9) display non-linear behavior (Fig. S1). Linear scaling patterns appear to display a single trend with little to no change in the relative swirling size. Non-linearly scaling ring patterns display at least a 40% increase in the relative swirling size from the start to the end of the time lapse. The maximum swirling size between linear (29.2 ± 2.4 μm) and non-linear (40.0 ± 3.8 μm) groups was also seen to be significantly different (p <0.001, t-test). The swirling size was also found to be significantly different between linear ring pattern and un-patterned monolayer (p < 0.001). Furthermore, direct calculation of energy dissipation from velocity shear rate showed agreement with the observed enstrophy trends, indicating that the phenomenon is consistent (Fig. S2). The non-linear energetics changes seen with time are preserved when plotted as a function of cell density (Fig. S3). The non-linear change in kinetic energy typically occurs around ~1.8 x 105 cell/cm2 in the non-linearly scaling ring patterns. Cell density of both linear and non-linear ring patterns share similar ranges, indicating that this is not a cell density-dependent behavior (Fig. S4). Time averaging of all linear and non-linear patterns shows that non-linear behavior is maintained (Fig. S5). However, there no longer appears to be a specific time point in which kinetic energy increases (Fig. S5 C). Longer imaging times of up to 48 hours after seeding showed that no further changes in scaling behavior occur once the new regime is established (Fig. S6). Finally, an examination of C2C12 cell behavior on ring patterns displayed linear behavior, indicating that the observed nonlinear scaling behavior may be specific to epithelial cells (Fig. S7).

3.2. Non-linear increase in kinetic energy is associated with coordinated motion

To elucidate the source of this non-linear scaling behavior, we examined the migration profiles and trends within individual ring patterns at the later time points (hour 24-36). The individual plot of kinetic energy and enstrophy for the linear scaling ring pattern show trends that decrease at proportional rates (Fig. 3 A). With the decreasing trend in kinetic energy, the change in the tangential velocity on the ring pattern is examined (Fig. 3 B). The tangential velocity on both the inner (red) and outer (blue) boundaries appears to start moving in opposing directions, as would be expected from normal chiral motion. At later times in the video, the motion at both boundaries appears to come to an arrest altogether.

Figure 3: The increase in kinetic energy is associated with coordination of migration on the ring pattern.

Figure 3:

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(A) The individual kinetic energy and enstrophy plots from a linear scaling ring pattern show coordinated trends with standard error of the mean shaded. (B) The tangential velocity profile of cells on the ring pattern shows that motion on the inner boundary (red, 100μm), outer boundary (blue, 300μm), as well as the average motion (green) all come to arrest at later time points. (C) A non-linear scaling ring pattern shows that at later time points kinetic energy increases while enstrophy continues to decrease with standard error of the mean shaded. (D) The tangential velocity profile of the non-linear pattern shows that the inner (red), outer (blue), and average (green) motion all start to increase at the same time. (E) The vorticity map of a non-linear pattern shows that at the start of the time lapse there is considerable randomness in swirling behavior. (F) At later time points from a non-linear pattern, there is less swirling behavior as velocity vectors all point in the same tangential direction. (G) The circulation of the monolayers overall highlights that non-linear patterns (magenta) see a consistent rotation direction of the entire ring while linear patterns (green) do not. (all scale bars = 50 μm, all arrows = 50 μm/hr).

In the non-linear scaling example, the plot of enstrophy is similar to that seen in the linear case (Fig. 3 C). However, the kinetic energy of the system increases at roughly 32 hours after seeding. The tangential velocity profile of the non-linear ring shows that throughout the time lapse, the direction of the outer boundary remains consistent, with the inner boundary resting closer to 0 μm/hr and then increasing towards the end (Fig. 3 D). Of the 9 non-linear ring patterned videos 55% indicate that there is even a change in the direction of the inner boundary migration direction (Fig. S1). This highlights that the outer boundary migration direction may dominate the inner boundary migration, and the entire ring-shaped tissue movement can be dragged by the motion of the cells on the outer boundary (Rahman et al., 2023). Like the averaged kinetic energy described in Fig. S5 C, averaging of absolute tangential velocity does not show a drastic increase at a specific time point (Fig. S8). Despite the variation between groups, an increase in kinetic energy associated with the coordination of migration on the inner and outer boundaries of the ring pattern can be seen. With this increasing velocity coordination, a decrease in enstrophy is additionally evident.

The vorticity map at the start of the time lapse shows a certain degree of randomness to the motion (Fig. 3 E). When there are local (1-2 cell length scale) increases in motion, there are what appear to be small eddy vortices that form on either side of the occurrences. Specifically, positive (blue) counterclockwise swirling groups of arrows are almost always accompanied by negative (red) clockwise swirling. In squaring the vorticity, the enstrophy term encapsulates the total swirling behavior in the monolayer. After 12 hours, the non-linear ring pattern video shows that all the velocity vectors point in roughly the same tangential direction, and there are very few of these eddy vortices formed (Fig. 3 F). This explains why the enstrophy of the system overall can continue to decrease while the kinetic energy continues to increase. To evaluate the large-scale cellular motion, we furthermore quantified the circulation of individual cell patterns, which is the summation of vorticity in the entire pattern. The circulation of the system shows that the non-linear ring pattern (magenta) continues to have a net rotation in the same direction, while the linear ring pattern (green) has an overall circulation that decreases towards 0 hr−1 (Fig. 3 G). For the nonlinear case, the net positive circulation increases 32 hours after seeding, consistent with the tangential velocity profile of the entire ring pattern.

3.3. Non-linear energetic scaling behavior is associated with chiral cell alignment

Since the non-linear increase in kinetic energy is associated with the rotation of the entire ring pattern in the same direction, we would like to see how this agrees with the chiral alignment of cells. In our previous work, we have seen that ring-patterned monolayers of cells usually see bidirectional migration on the inner and outer boundaries (Wan et al., 2011). This is in contrast to the uni-directional migration seen here at later time points (Fig 3. D). For each video, the mean angle of alignment with respect to the center of the ring pattern was determined (Fig. 4 A–D). Since non-linear behavior was observed primarily in the later stages of the video, only the alignment during the last 4 hours was considered (24 frames, 10 minutes between frames). Using a rank test, a video was be considered chiral if greater than 70% (or more than 17) frames were determined to be chiral (p < 0.05). Of the linearly scaling ring patterns, only 18% (3) were found to be significantly chiral (Fig. 4 E). Non-linear rings showed that 89% (8) of ring patterns had significant chirality. By the proportionality test, the ratio of chiral and non-linear scaling rings is significantly different from chiral and linear scaling rings (p<0.001). This shows that non-linear ring patterns overall see a large degree of chiral alignment while they rotate in a coordinated fashion, indicating cell chirality may contribute to the non-linear energetic behavior of patterned monolayers.

3.4. Reduction of cell-cell adhesion abrogates non-linear scaling behavior

When non-linear scaling behavior occurs, it appears that the outer boundary dominates the behavior of the ring and eventually drags the inner boundary along with it (Fig. 3 D). If the outer boundary is dominant, it would suggest that cells there exerts some force on those in the inner regions through cell-cell adhesions. To decrease cell-cell adhesion of the monolayer, cells were treated with 0.5 and 1.0 mM EGTA, which acts as an extracellular calcium chelator and decreases E-cadherin activity (Rothen-Rutishauser et al., 2002). Overall, treated ring patterns produced mainly linear energetic scaling behavior (Fig. 5 A and B). Only two ring patterns treated with 0.5mM EGTA showed at least a 40% change in the relative swirl size (non-linear behavior). The patterns treated with 1.0 mM EGTA showed no non-linear behavior at all (Fig. 5 C). Proportionality testing indicated that 0.5 mM EGTA rings and untreated rings show significantly different ratios of linear and non-linear behavior. The difference between 1.0 mM and 0.5 mM EGTA treated groups was found to not be significant. This highlights that cell-cell adhesion may be necessary for generating non-linear energetic scaling through migration coordination.

Figure 5: Disruption of cell-cell adhesion through EGTA treatment reduces non-linear scaling behavior.

Figure 5:

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(A-B) Examples of linear energetic scaling from ring-patterned monolayers treated with 0.5mM and 1.0mM EGTA (respectively) 20 hours after seeding. (C) The distribution of non-linear scaling events shows that the behavior has been abolished altogether by 1.0 mM EGTA treatment, while it can still be seen in the 0.5 mM treated group.

4. DISCUSSIONS

In this work, we found that the coordination of epithelial monolayers was associated with non-linear scaling of kinetic energy and enstrophy. Monolayers of un-patterned and ring-patterned MDCK cells were analyzed using PIV to infer the direction and speed of local migration. It was seen that un-patterned monolayers and some patterned monolayers displayed a linear trend of kinetic energy and enstrophy. Other ring patterns showed an increase in kinetic energy at later time points that were not adjoined by a concurrent increase in enstrophy. Unique non-linear scaling behavior was specifically seen to correlate with coordinated motion across the entire pattern as well as with chiral cellular alignment. Once coordinated motion is developed, there appears to be a decrease in swirling behavior at the 1-2 cell length scale. Disruption of coordination by drug treatment saw a decrease in the number of patterns that displayed this non-linear scaling behavior. A schematic of the change in small-scale swirling to large-scale coordination illustrates the biophysical processes accompanying the non-linear energetic trend observed (Fig. 6).

Figure 6: Schematic detailing the physical changes that correlate with the energetic behavior of the patterned monolayers.

Figure 6:

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The vorticity in the counterclockwise (blue) or clockwise (red) directions is indicated by color. The left portion of the figure indicates linear behavior, where monolayer motion is largely random with a lot of small-scale vorticity behavior. The right side shows non-linear scaling behavior where the motion of cells from the outer boundary to the inner is coordinated, leading to more homogenous vorticity.

Changing migration patterns that are seen appear to later coincide with the direction that the outer boundary migrates. This implicates significant cell-cell interactions in aiding the coordination of cell migration. Patterns of migration seen here is reminiscent of the several different phenomena seen in fluid flow of highly viscous material. MDCK cells seeded onto circular islands have the ability to generate coordinated flow about the center of the pattern (Doxzen et al., 2013; Luo et al., 2023). Flow and alignment about the center of the pattern becomes reminiscent of a topological defect. The impact of defects in cellular alignment within monolayers has been extensively studied. Many studies have examined the alignment and morphological events surrounding topological defects such as extrusion (Guillamat et al., 2022; Hoffmann et al., 2022; Saw et al., 2017) and modulations in monolayer density (Balasubramaniam et al., 2021; Kawaguchi et al., 2017; Zhang et al., 2021). In this study, we see that non-linear energetic scaling is associated with the coordinated motion about the center of the micropattern, reminiscent of an integer defect. This behavior has been described as solid rotation with correlation of motion at significant length scales on circular micropatterns (Doxzen et al., 2013). Additionally, the reduction of cell-cell interactions sees less coordination of monolayer migration. Here, we see very similar behavior, with the outer boundary of the pattern appearing to dominate. Differences observed between linear and non-linear modes of micropattern migration indicate that there may be a significant change in the shear behavior within the monolayer. This is indicated by the similar trends observed between enstrophy and energy dissipation which are directly calculated from vorticity and the velocity shear rate (Fig. S2). Reduction of small-scale vorticity behavior supports the belief that the monolayer is experiencing differential shear stress (Fig. 2 E and F). Once cells coordinate their motion and all velocity vectors point in the same direction, there are fewer eddy-like vortical events. Less vorticity within the monolayer would indicate there is less sliding occurring between cells (i.e. less shear).

Non-linear behavior here overall highlights the transition of the monolayer towards a solid-like state while maintaining motion. Changing motion from solid to fluid-like behavior displays similarities to glassy transitions seen in other active materials (Berthier et al., 2019). Previous work from our lab has further described the importance of jamming transitions on ring patterns (Rahman et al., 2023). Both linear and non-linear energetic scaling ring patterns experience these transitions. Linear patterns transition from a turbulent fluid motion towards an overall arrest in motion, i.e. a solid at rest. Non-linear patterns are unique in that they maintain motion after the solid transition occurs. It is not clear if this avoidance of arrest in motion serves a specific purpose for the development of tissue function and morphology in the presence of geometric constraints. Since we see that chiral alignment is associated with the non-linear scaling behavior, it is possible that this increase in motility plays a role in generating asymmetric morphology in some cases.

Further work is required to examine the association of chirality, glassy transitions, and non-linear scaling behavior. Since non-linear scaling does not appear to be associated with time or density directly, more work is specifically needed on what biomechanical cues could lead to the generation of this coordinated migration dependent behavior and how chirality may play a more active role. Significant work has already been completed in regards to modeling the behavior in epithelial monolayers (Rahman et al., 2023; Yamamoto et al., 2020). However, more work is needed to understand how this affects the development of monolayers of different cell types (endothelial, fibroblasts, etc.). To ensure the reduction of cell-cell activity, knockdown of adhesion molecules (such as E-cadherin) may provide a more robust change in migration. Furthermore, the energetic development of monolayers in subregions surrounding naturally occurring forms of geometry and deformation (i.e. topological defects) would provide useful information on how they can affect the morphology of tissues in vivo.

5. CONCLUSION

In summary, we found that the coordinated motion of epithelial micropatterned monolayers generates the unique non-linear energetic scaling behavior. Migration coordination across the inner and outer boundaries of the pattern produces an increase in the kinetic energy of the system while the small-scale vorticity behavior subsides. The change in ring-patterned monolayer behavior was further found to be associated with chiral cell alignment and cell-cell adhesion. Overall, this study highlights a unique energetic scaling phenomenon of developing epithelium within confined geometries that has not been reported before.

Supplementary Material

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ACKNOWLEDGMENTS

This work was supported by the National Institutes of Health (HL148104 and GM141387 to LQW). FDP and HZ were supported by the American Heart Association Predoctoral Fellowship 23PRE1026748 and 20PRE35210243, respectively.

Footnotes

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CRediT authorship contribution statement

Frank D. Peters: Writing – review & editing, Writing – original draft, Methodology, Conceptualization, Investigation. Tasnif Rahman: Methodology, Writing - original draft. Haokang Zhang: Methodology, Writing - original draft. Leo Q. Wan: Writing – review & editing, Validation, Funding acquisition, Conceptualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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