⚛️ biophysics

A mechanical model of multicellular remodelling in epithelial monolayers

This paper introduces a Dynamic Reference Frame (DRF) within an off-lattice cell-centre model to simulate subcellular remodelling in epithelial monolayers, enabling the reproduction of multiscale mechanical responses and the analysis of how deformation history influences tissue recovery.

Original authors: Osborne, J. M., Van Ammers, R., Davit, Y., Gavaghan, D. J., Byrne, H.

Published 2026-09-14

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CC BY 4.0

Original authors: Osborne, J. M., Van Ammers, R., Davit, Y., Gavaghan, D. J., Byrne, H.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

The outer layers of our bodies and the linings of our internal organs are made of epithelial tissues, sheets of cells that act as protective barriers and gatekeepers for the body. These tissues are not static; they constantly reshape themselves, stretch, and relax as organs grow, heal, or respond to their environment. Scientists have long known that when these sheets are pulled or pushed, they do not simply snap back or stretch out in a single, uniform motion. Instead, they react in two distinct phases: a quick, elastic snap that happens in seconds, followed by a much slower, creeping adjustment that can take minutes or even hours. This slow change is a form of remodeling, where the internal machinery of the cells—specifically the protein scaffolds and the sticky bonds holding them together—reorganizes to accommodate the new shape. Understanding this two-speed behavior is crucial because it reveals how tissues maintain their integrity under stress and how they might fail in disease, yet existing computer models have struggled to capture this complexity without becoming impossibly complicated.

A team of researchers has now developed a new way to simulate this behavior, creating a digital model that captures both the fast snap and the slow creep of epithelial sheets. Rather than trying to model every tiny protein inside each cell, which would require immense computing power, the scientists introduced a concept they call a "dynamic reference frame." Imagine the tissue as a collection of cells connected by springs. In older models, the length of these springs was fixed; if you pulled the tissue, the springs stretched and then held that new length forever unless the cells moved apart. This new model adds a second, invisible layer of springs that represents the cell's internal memory of its shape. When the tissue is stretched, the visible cells move quickly, but this invisible layer slowly shifts its own target length to match the new reality. This shift represents the biological process where cells reorganize their internal structures to adapt to the stress they are feeling.

The researchers tested this idea by running computer simulations of creep and stress relaxation experiments, which are standard ways to test how materials behave under force. In a creep experiment, a constant pull is applied to the edge of a digital tissue sheet, and the researchers watched how much it stretched over time. In a stress relaxation experiment, the tissue is stretched to a fixed width and held there while the force required to keep it stretched is measured. When the model included only the fast, elastic response, the tissue behaved like a simple rubber band: it stretched quickly and then stopped, or the force dropped quickly and then stayed steady. However, when the researchers turned on the dynamic reference frame to allow for subcellular remodeling, the simulation changed dramatically. The tissue still reacted quickly at first, but then it continued to stretch slowly over time, or the force continued to drop, perfectly mimicking the two-phase behavior seen in real biological experiments.

By adjusting the settings of this new model, the team discovered that the speed of this slow remodeling depends on how sensitive the cells are to stress and how quickly they update their internal "memory." They found that if the cells are slow to adapt, the tissue behaves mostly like a simple elastic material. But if the cells are sensitive and update their internal structure over time, the tissue exhibits the complex, multi-speed response observed in nature. Crucially, the model also showed that the history of the stress matters. If a tissue is held under tension for a long time, its internal structure adapts significantly. When the tension is finally released, the tissue does not snap back immediately; instead, it recovers slowly, and the rate of this recovery depends on how long it was held. This is a key finding because many older computer models predicted that the recovery would be the same regardless of how long the tissue was stretched, a result that contradicts real-world observations.

The study confirms that the slow, creeping changes in tissue shape are driven by the continuous reorganization of the cells' internal components, rather than by cells dividing or moving to new positions. The researchers demonstrated that this behavior can be captured without needing to simulate every single molecule inside a cell, making it possible to study large sheets of tissue containing thousands of cells. Their work suggests that the ability of tissues to remember and adapt to past stresses is a fundamental property of their mechanical design. By providing a framework that links the microscopic reorganization of cells to the macroscopic behavior of tissues, this new model offers a clearer path to understanding how organs develop, how they heal, and how they might respond to the mechanical forces of disease. The code and data used to generate these findings are now available for other scientists to use, allowing the community to build more detailed and accurate models of living tissues.

Technical Summary: A Mechanical Model of Multicellular Remodelling in Epithelial Monolayers

Problem Statement
Epithelial monolayers, which form the basis of many mammalian organs, exhibit complex mechanical responses to external forces. Experimental evidence (e.g., Harris et al. [6]) demonstrates that these tissues respond to deformation across multiple timescales: a fast elastic response (seconds) followed by slower relaxation or creep (minutes to hours). Existing mathematical models struggle to reproduce this behavior without invoking cell proliferation or rearrangement, which are negligible on these short timescales.

Continuum models (e.g., Standard Linear Solid) typically assume a single characteristic timescale.
Complex viscoelastic models (e.g., multiple Maxwell elements) can produce multiple timescales but are often phenomenological, with parameters difficult to link to specific subcellular mechanisms.
Cell-based models (e.g., Vertex Dynamics, Overlapping Spheres) generally rely on fixed interaction laws. When proliferation and rearrangement are neglected, they also tend to exhibit single-timescale responses. While Subcellular Element (SE) models can capture multi-timescale behavior via internal reorganization, their high computational cost limits them to very small systems (<10 cells), making them impractical for tissue-scale simulations.

Methodology
The authors propose a novel extension to off-lattice, cell-centre models, specifically utilizing a Voronoi Tessellation (VT) framework. The core innovation is the introduction of a Dynamic Reference Frame (DRF).

Model Architecture: The tissue is represented by three sets of cell centers:
1. Physical cells ( $r_i$ ): The actual, deforming positions of the cells.
2. Reference cells ( $\rho_i$ ): A dynamic configuration that evolves over time, representing the "remodeled" state of the tissue.
3. Static cells ( $s_i$ ): A fixed initial configuration serving as a baseline.
Governing Equations:
- Physical Motion: Physical cells move according to a force balance between viscous drag, external forces, and spring forces connecting physical cells to the reference cells (Eq. 2). The spring force depends on the difference between the physical distance ( $||r_{ij}||$ ) and the reference distance ( $||\rho_{ij}||$ ).
- Reference Evolution: The reference cells evolve according to a separate equation of motion (Eq. 3). They are driven by two competing forces:
  1. A restoring force pulling them toward the static configuration ( $s_i$ ).
  2. A remodelling force pulling them toward the time-averaged physical configuration ( $\bar{r}_{ij}$ ) over a memory window $T_M$ .
Non-dimensionalization: The system is non-dimensionalized to define key parameters:
- $\alpha$ (Remodelling ratio): The relative strength of the remodelling force vs. the restoring force.
- $\eta$ (Remodelling rate): The rate at which the reference frame evolves relative to the physical frame.
- $T_M$ (Remodelling memory): The time window over which the reference frame averages physical deformation.
Simulation Setup: The model simulates 2D epithelial monolayers (105 cells) under two experimental protocols:
1. Creep Relaxation: Constant force applied; strain is recorded.
2. Stress Relaxation: Strain is ramped to a maximum and held; stress is recorded.
Analysis: Time-series data are fitted with sums of exponentials. The number of timescales is determined by whether a single exponential provides a parsimonious fit (error $E_1 < 0.05$ ) or if multiple terms are required.

Key Contributions

Dynamic Reference Frame (DRF): The paper introduces a mechanism to decouple the instantaneous deformation of a cell from its mechanical reference configuration. This allows the model to simulate subcellular remodelling (cytoskeletal reorganization, myosin turnover, adhesion bond remodelling) without explicitly modeling subcellular components, thus avoiding the computational cost of SE models.
Multi-timescale Emergence: The authors demonstrate that the DRF framework naturally generates multi-timescale responses (fast elastic + slow relaxation/creep) in cell-based models, a feature previously absent in standard VT or cell-centre models without proliferation.
Parameter Sensitivity: The study quantifies how the remodelling parameters ( $\alpha, \eta, T_M$ ) control the transition between single- and multi-timescale behaviors.
History Dependence: The model successfully reproduces the experimental observation that tissue recovery upon release depends on the duration of the imposed strain, a phenomenon standard elastic models fail to capture.

Results

Creep Experiments:
- Without remodelling ( $\alpha=0$ ), the tissue exhibits a single-timescale response (fast elastic deformation followed by a plateau).
- With remodelling ( $\alpha > 0$ ), a second, slower timescale emerges as the reference frame evolves, allowing further deformation.
- The emergence of multiple timescales is robust across applied force magnitudes but is highly sensitive to $\alpha$ (remodelling strength) and $\eta$ (remodelling rate). Increasing the memory window ( $T_M$ ) also promotes multi-timescale behavior.
Stress Relaxation Experiments:
- Similar to creep, remodelling introduces a secondary relaxation phase.
- The ramp time ( $T_R$ ) acts as a control parameter; intermediate ramp times expose distinct post-ramp relaxation timescales, while very short or very long ramps may mask them.
- Increasing $T_M$ expands the parameter space where multi-timescale responses are observed.
Recovery from Strain:
- In models without remodelling, the rate of recovery after releasing a held strain is independent of the hold duration ( $T_H$ ).
- In the DRF model, the recovery rate is inversely proportional to $T_H$ . The model predicts that longer hold times allow the reference frame to deform further (representing accumulated subcellular remodelling), resulting in lower stress at release and slower subsequent recovery. This behavior aligns with the general experimental finding by Harris et al. [6] that recovery depends on deformation duration, while the specific $1/T_H$ relationship is a quantitative result of the proposed model.

Significance and Claims
The paper claims to provide a computationally efficient framework for studying subcellular remodelling at the tissue scale. By extending standard cell-centre models with a DRF, the authors can reproduce complex, multi-timescale mechanical behaviors observed in epithelial monolayers without resorting to prohibitively expensive subcellular simulations or phenomenological continuum models with unmeasurable parameters.

The authors state that this framework allows for "more detailed descriptions and analyses of the development and function of biological tissues." They emphasize that the model is modular: the DRF concept can be applied to other off-lattice models (e.g., Vertex Dynamics) and extended to 3D or to include processes like cell growth and differentiation by adding intermediate reference states. The work suggests that the history of deformation, mediated by subcellular turnover, is a critical factor in tissue mechanics that can now be systematically investigated using this approach.

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