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Mean-field interactions between living cells in linear and nonlinear elastic matrices

This paper investigates how living cells mechanically interact within linear and nonlinear elastic matrices by calculating the work required to deform the surrounding medium, revealing that interaction energy depends on cell geometry and material properties in linear matrices, while in strain-stiffening nonlinear matrices, cell contraction is limited by diverging shear stress.

Original authors: Chaviva Sirote, Yair Shokef

Published 2026-08-27

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CC BY 4.0

Original authors: Chaviva Sirote, Yair Shokef

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 cells are not merely passive passengers drifting through the body; they are active engineers that constantly push and pull against the soft, gel-like material surrounding them. This material, known as the extracellular matrix, acts as a scaffold, but it also serves as a communication network. When a cell contracts, it creates tiny ripples of stress and strain that travel through this gel, allowing distant cells to feel one another's presence without ever touching. This mechanical dialogue is vital for life, guiding how stem cells turn into specific tissues, how wounds close, and how embryos take shape. Scientists have long understood that these forces exist, but the precise rules governing how cells interact through this elastic web—especially when the material itself changes its properties under pressure—have remained difficult to pin down.

A team of researchers at Tel Aviv University has now mapped these interactions with new clarity, focusing on how the stiffness of the surrounding environment alters the relationship between neighboring cells. They investigated two distinct ways cells might behave: some cells act like rigid rulers, insisting on shrinking by a specific amount regardless of the resistance they face, while others act like constant-pressure pumps, applying a fixed force and letting the resulting movement vary. By building mathematical models of cells as spheres and cylinders embedded in elastic materials, the team calculated the mechanical work required for these cells to deform their surroundings. Their work reveals a fundamental rule: cells that control their shape repel one another, while cells that control their force are drawn together.

The researchers began by examining a world where the surrounding material behaves predictably, like a standard rubber band that stretches in direct proportion to the pull. In this linear world, they found that the distance between cells matters deeply. When a cell tries to shrink by a fixed amount, the presence of a neighbor makes the job harder. The neighbor acts as a barrier, forcing the cell to push harder against the material to achieve the same contraction. This extra effort represents a positive energy cost, which translates into a repulsive force between the cells. Conversely, if a cell applies a fixed amount of force, the presence of a neighbor makes it harder for the cell to move. The cell ends up shrinking less than it would in isolation, and this reduction in movement lowers the total energy of the system, creating an attractive pull. The study confirmed that in these simple, linear materials, the strength of this interaction depends on the distance between the cells and the stiffness of the material, but not on the internal stiffness of the cell itself when the cell is controlling its shape.

However, the real world is rarely so simple. The biological gels that surround cells are not uniform rubber bands; they are complex networks of fibers that become significantly stiffer when stretched or sheared. This phenomenon, known as strain stiffening, means that the material resists deformation more and more fiercely as the force increases. The researchers extended their models to include this nonlinear behavior, simulating how cells interact in these more realistic, toughening environments. They discovered a dramatic limit to how much a cell can contract when surrounded by others in such a material. As the cell tries to shrink, the shear stress in the surrounding gel rises sharply. Because the gel stiffens under this stress, it eventually becomes so resistant that the cell cannot shrink any further, no matter how much force it applies. The researchers found that there is a maximum possible contraction for a cell in a crowded, nonlinear environment, determined by the distance to its neighbors and the material's inherent limits.

In these nonlinear scenarios, the interaction energy behaves differently than in the simple linear case. For cells that try to maintain a fixed shape, the energy required to contract grows rapidly and eventually becomes infinite at that maximum limit, effectively locking the cells in place. For cells that apply a fixed force, the interaction remains attractive, but the relationship between force and movement becomes highly complex. The team's simulations showed that even in these chaotic, stiffening environments, the basic rule holds: shape-regulating cells repel, and force-regulating cells attract. The study suggests that the mechanical cost of moving through a stiffening gel is a primary driver of cell behavior, potentially explaining why cells in crowded tissues might stop moving or align themselves in specific patterns. By treating the cell as a passive elastic object and focusing on the mechanics of the surrounding medium, the researchers provided a clear, quantitative picture of how the physical world dictates the social behavior of living cells.

Technical Summary: Mean-field interactions between living cells in linear and nonlinear elastic matrices

Problem Statement
Living cells embedded in or adhered to an elastic environment transmit mechanical forces through the deformation of the surrounding medium. These matrix-mediated interactions influence critical biological processes, including stem-cell differentiation, wound healing, and cancer metastasis. While previous theoretical work has modeled cell contraction as active force dipoles in linear elastic media, there is a need to understand how these interactions depend on cell geometry (beyond simple spheres), the mechanical properties of the cell interior, and the nonlinear, strain-stiffening nature of biological extracellular matrices (ECMs). Specifically, the paper investigates how the mechanical work performed by a cell changes when it is surrounded by other cells, distinguishing between cells that regulate their surface displacement and those that regulate their active stress.

Methodology
The authors employ a mean-field approach to model the spatial arrangement of cells. Instead of simulating a complex array of cells, they model a single contracting cell surrounded by a "rigid wall" at a distance $R_1$ , representing the symmetry surface between neighboring cells. Two geometries are analyzed:

Spherical Symmetry: A cell of radius $R_0$ in a 3D matrix.
Cylindrical Symmetry: A cell of radius $R_0$ in a 3D matrix, considering both plane strain (axial constraint) and plane stress (free axial deformation) conditions.

The study considers two regulatory behaviors:

Displacement Regulation: The cell generates a fixed surface displacement $u_0$ regardless of the environment.
Stress Regulation: The cell applies a fixed active stress $\sigma_0$ regardless of the resulting displacement.

The analysis is divided into two regimes:

Linear Elasticity (Section III): Analytical solutions are derived for the displacement field and interaction energy ( $\Delta E$ ) in linear media, accounting for the shear ( $\mu$ ) and bulk ( $K$ ) moduli of both the cell ( $c$ ) and the matrix ( $m$ ).
Nonlinear Elasticity (Section IV): The matrix is modeled as a strain-stiffening material using a specific energy density function where the shear modulus increases with stress ( $G \propto \sigma^{3/2}$ at high stress). The governing nonlinear ordinary differential equations are solved numerically for the spherical geometry, with analytical approximations derived for the near-field and far-field behaviors.

Key Contributions and Results

Linear Elastic Regime:
- Displacement Regulation: The interaction energy is positive (repulsive). For spherical geometry, the interaction depends only on the matrix moduli ( $K_m, \mu_m$ ) because the cell deformation is pure compression. For cylindrical geometry, the interaction also depends on the matrix moduli but the decay rate differs ( $1/R_1^2$ vs $1/R_1^3$ for spheres).
- Stress Regulation: The interaction energy is negative (attractive). In spherical geometry, the interaction depends on the cell's bulk modulus ( $K_c$ ) but not its shear modulus, as the cell deformation remains pure compression. In cylindrical geometry, the interaction depends on both the cell's bulk and shear moduli because the deformation includes shear.
- The results confirm that the sign of the interaction is determined by the regulatory strategy: displacement regulation leads to repulsion, while stress regulation leads to attraction.
Nonlinear (Strain-Stiffening) Regime:
- Limit on Contraction: In a strain-stiffening matrix, the shear stress diverges as the strain approaches a maximal value $\gamma_a$ . Consequently, for a bounded geometry (finite cell-cell distance $R_1$ ), there exists a maximal possible cell contraction ( $u_a$ ). If a cell attempts to contract beyond this limit, the shear stress required to satisfy the zero-displacement boundary condition at $R_1$ becomes infinite.
- Scaling Laws: The maximal contraction $u_a$ scales linearly with the material's maximal shear strain $\gamma_a$ and logarithmically with the dimensionless cell-cell distance ( $R_1/R_0$ ).
- Interaction Energy Divergence: Under displacement regulation, the interaction energy diverges as the cell contraction approaches the maximal limit $u_a$ . Under stress regulation, the interaction energy remains finite and scales quadratically with stress ( $\Delta E \propto \sigma_0^2$ ), even in the highly nonlinear regime, because the cell can generate arbitrarily large stresses without reaching the divergence point of the displacement.

Significance and Claims
The paper claims to provide a comprehensive analytical and numerical framework for understanding matrix-mediated cell-cell interactions in both linear and nonlinear environments. Its primary significance lies in:

Extending Geometry: Moving beyond spherical symmetry to include cylindrical setups relevant to cells on substrates or elongated cells (e.g., neurons).
Incorporating Cell Stiffness: Explicitly including the elastic rigidity of the cell interior, showing that it influences interaction energy only under stress regulation and in non-spherical geometries.
Nonlinear Limits: Demonstrating that in strain-stiffening matrices, the presence of neighboring cells imposes a fundamental limit on how much a cell can contract if it regulates displacement. This suggests that the mechanical cost of contraction becomes prohibitive as cells approach each other in stiffening gels.
Regulatory Mechanisms: Reinforcing the theoretical prediction that the sign of mechanical interaction (attraction vs. repulsion) is dictated by whether cells regulate displacement or stress, a finding that holds true even in complex nonlinear media.

The authors note that while they treat the cell as a passive linear solid (or a soft inclusion in the nonlinear case), the results for displacement regulation are robust regardless of the cell's internal complexity, as the cell's interior deformation remains identical in both bounded and unbounded scenarios. The work serves as a theoretical basis for understanding how mechanical feedback loops might drive cell alignment, densification, or migration in biological tissues.

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