🔬 condensed matter

Quantifying cell shape and density fluctuations in epithelial tissue in vivo

This study presents a theoretical framework modeling epithelial cells as interacting soft ellipsoids to derive continuum stochastic equations that, when validated against *Drosophila* pupal wing data, quantify critical parameters like effective tissue stiffness and active cell shape changes through spatial-temporal correlation functions.

Original authors: Jake Turley, Helen Weavers, Paul Martin, Isaac V. Chenchiah, Tanniemola B. Liverpool

Published 2026-09-10

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CC BY 4.0

Original authors: Jake Turley, Helen Weavers, Paul Martin, Isaac V. Chenchiah, Tanniemola B. Liverpool

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Living tissues are not static bricks in a wall; they are dynamic, breathing communities where individual cells constantly shift, stretch, and squeeze against one another. To understand how a body grows or heals, scientists must look beyond the average shape of a cell and pay attention to the tiny, fleeting deviations from that average. These fluctuations, the momentary wobbles and stretches of individual cells, carry hidden information about the tissue's mechanical health, much like the way the sway of a crowd can reveal the stiffness of the ground beneath their feet. By measuring these subtle movements, researchers can deduce the effective "stiffness" of a tissue and how actively its cells are working to change their form, offering a non-invasive window into the physical forces that drive development and repair.

In a study focused on the developing wing of a fruit fly, researchers have built a new theoretical framework to capture these dynamics in living tissue. They chose the pupal wing of the fruit fly Drosophila, a flat, single layer of cells that stretches and reshapes itself as the insect matures. Using high-powered microscopes and advanced image analysis, the team tracked the shapes and positions of over a million individual cells over a three-hour period. They observed that while the cells generally elongated in a specific direction due to the tension of the surrounding tissue, each cell also exhibited its own unique, random fluctuations in shape and size. To make sense of this chaos, the scientists modeled the cells as soft, squishy ellipses that interact with their neighbors, allowing them to change their area, stretch, and rotate. By averaging the behavior of these millions of cells, they derived a set of equations that describe how local deviations in cell shape and density spread through the tissue over time and space.

The researchers then compared the predictions of their mathematical model against the actual data collected from the fly wings. They found that the model could accurately reproduce the patterns of how cell shapes correlated with one another across different distances and time intervals. This allowed them to extract specific numerical values that represent the physical properties of the tissue. They determined a value for the tissue's effective "stiffness," which describes how resistant the tissue is to deformation, and a measure of "active shape change," which quantifies how vigorously the cells are remodeling themselves. The study revealed that the tissue behaves like a material with a specific rigidity, but one that is also driven by internal, active forces generated by the cells themselves. The model showed that the movement of cells relative to their neighbors was surprisingly small, suggesting that the tissue maintains a tight, cohesive structure while still allowing for the necessary reshaping required for development.

To test if this framework could detect changes in tissue health, the scientists applied their method to wounded wings. They created small, circular wounds in the tissue and watched as the cells closed the gap. Once the wounds had sealed, they measured the tissue properties again. They found that the tissue near the healed wound had become softer, with a lower effective stiffness, which likely made it easier for the cells to deform and close the injury. More importantly, they observed a significant increase in the rate of active shape changes in the cells near the wound, indicating that the cells were working harder to remodel themselves to facilitate healing. This contrasted sharply with the behavior of genetically modified flies where a specific signaling pathway was blocked. In these modified flies, the cells failed to increase their active shape changes after wounding, and the tissue remained softer for longer. This lack of active remodeling correlated with the known delay in wound healing for these flies, confirming that the model could distinguish between passive mechanical softness and the active biological effort required for repair.

The study demonstrates that by listening to the quiet fluctuations of cells, scientists can quantify the mechanical state of a living tissue without disturbing it. The researchers showed that they could separate the passive, structural properties of the tissue from the active, energy-driven behaviors of the cells. This approach offers a powerful tool for understanding how tissues respond to injury or genetic changes, providing a way to measure the "stiffness" and "activity" of a tissue in real time. The ability to distinguish between a tissue that is simply soft and one that is actively trying to change its shape could help researchers understand the physical mechanisms behind complex biological processes, from the formation of organs to the body's response to injury.

Technical Summary: Quantifying Cell Shape and Density Fluctuations in Epithelial Tissue In Vivo

Problem Statement
Epithelial tissues are heterogeneous systems where local deviations from average cell shape and size (fluctuations) play a critical role in biological processes such as development and wound healing. While existing models either simulate individual cells (e.g., Vertex or Cellular Potts models) or use continuum equations that ignore cellular-scale information, there is a need for a framework that bridges these scales. Specifically, the authors aim to quantify how local fluctuations in cell shape and density evolve over time and how these fluctuations can reveal macroscopic mechanical properties, such as effective tissue "stiffness" and active cell remodeling, within the developing Drosophila pupal wing.

Methodology
The study combines high-resolution in vivo imaging with a theoretical reductionist framework:

Experimental Data Acquisition:
- System: The study focuses on the Drosophila pupal wing epithelium from 18 to 21 hours after puparium formation (APF), a period characterized by hinge contraction, wing blade stretching, and cell elongation along the proximal-distal (P/D) axis.
- Imaging: Confocal microscopy was used on pupae expressing E-cadherin-GFP (cell boundaries) and Histone2-RFP (nuclei). Images were captured every 2 minutes over a 3-hour window.
- Analysis: A deep learning algorithm detected cell boundaries, which were fitted with polygons to extract cell area and shape. Cell shapes were encoded using a second-rank $q$ -tensor, representing deviations from circularity (elongation and orientation). Over 1,000,000 cell-level measurements were collected.
Theoretical Framework:
- Microscopic Model: Cells are modeled as interacting soft ellipsoids with varying size and aspect ratio. The system is governed by stochastic differential equations (SDEs) for cell positions ( $r_i$ ) and shape tensors ( $q_i$ ). These equations include deterministic terms (active alignment, pairwise interactions via a potential $U$ ) and stochastic noise terms (representing mechanical and chemical fluctuations).
- Coarse-Graining: The microscopic equations are coarse-grained to derive continuum stochastic differential equations for the local density field ( $\hat{\rho}$ ) and the local shape field ( $\hat{Q}$ ).
- Linearization: The equations are linearized around mean values, separating the dynamics into "annealed" (time-dependent) fluctuations and "quenched" (time-independent structural) deviations.
- Correlation Functions: The authors derive analytical expressions for spatio-temporal correlation functions of density ( $C_{\rho\rho}$ ) and shape ( $C_{qq}$ ) fluctuations. These functions depend on parameters representing diffusion, active shape changes, and effective tissue stiffness.

Key Contributions and Results
The primary contribution is a theoretical framework that successfully fits experimental correlation functions to extract quantitative mechanical parameters from living tissue.

Parameter Extraction from Wild-Type Tissue:
By fitting the theoretical correlation functions to the experimental data, the authors determined several key parameters (Table I):
- Effective Stiffness ( $L$ ): $10.6 \pm 0.5 \, \mu\text{m}^4 \text{min}^{-1}$ . This parameter encodes the tissue's resistance to deformation.
- Active Shape Change ( $A$ ): $(2.83 \pm 0.48) \times 10^{-3} \, \text{min}^{-1}$ . This represents the rate of active relaxation of cell shapes toward a preferred state.
- Diffusion and Noise: The diffusion constant for density ( $D$ ) was found to be $7.4 \, \mu\text{m}^2 \text{min}^{-1}$ , significantly larger than the cell center diffusion ( $D_r$ ), suggesting collective motion dominates.
- Structural Deviations: The study identified a "quenched" structural density component with a decay length ( $\lambda$ ) of $23 \, \mu\text{m}$ , attributed to upstream developmental processes (e.g., vein formation).
Validation via Perturbation Experiments:
The framework was tested on perturbed tissues to verify its ability to distinguish mechanical states:
- Laser Wounding: Following wound closure, the tissue near the wound site exhibited a 22% reduction in effective stiffness ( $L$ ), suggesting the tissue becomes more deformable to facilitate healing. Conversely, the active shape change parameter ( $A$ ) increased by a factor of 1.6 near the wound, indicating accelerated active remodeling. Far from the wound, $A$ remained similar to healthy tissue, suggesting spatially restricted activity.
- Genetic Perturbation (JNK Signaling): In tissues where JNK signaling was inhibited (via bsk dominant negative expression), wound healing was delayed. The model revealed that while stiffness ( $L$ ) decreased (similar to wild-type wounded tissue), the active shape change parameter ( $A$ ) failed to increase. This lack of active remodeling correlates with the observed slower healing, demonstrating the model's ability to decouple passive stiffness from active biological drive.

Significance and Claims
The paper claims that this reductionist approach allows for the quantification of tissue-scale mechanical properties using only leading-order terms and a relatively small number of parameters. By fitting the model to experimental data, the authors demonstrate that:

Spatio-temporal correlation functions can serve as a non-invasive probe to measure effective tissue "stiffness" and active remodeling rates.
The framework can distinguish between passive mechanical changes (altered stiffness) and active biological responses (changes in remodeling rates) in response to perturbations like wounding or genetic inhibition.
The model provides a bridge between microscopic cell-cell interactions and macroscopic tissue behavior, offering a method to track how tissue properties evolve during morphogenesis and repair without the need for invasive laser-recoil experiments.

The authors note that while the model provides a "reasonably good fit" and captures the essential physics, it relies on simplifying assumptions (e.g., homogeneous stiffness, neglecting nematic symmetry breaking). They suggest that future refinements could include anisotropic stiffness parameters to better match the data, but the current framework is sufficient to extract meaningful effective parameters from complex in vivo systems.

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