Source-linked AI summary
Active Vertex Model for Cell-Resolution Description of Epithelial Tissue Mechanics
Daniel L. Barton, Silke Henkes, Cornelis J. Weijer, Rastko Sknepnek
TL;DR
The paper addresses the need for cell-resolution models that describe the mechanics and collective migration of large confluent epithelial tissues beyond quasi-static or computationally limited approaches. It introduces an Active Vertex Model combining Vertex Model mechanics with active matter dynamics and cell-centre-based Voronoi geometry. The model supports efficient large-system simulations, natural T1 transitions, and additional tissue processes, while the authors identify simplified activity and planar geometry as important limitations.
Problem
How to describe coordinated epithelial mechanics and migration at cell resolution across large tissues remains limited by quasi-static assumptions and computationally complex topology changes.
Method
The AVM combines Vertex Model mechanics with active matter dynamics, tracking cell centres whose Delaunay triangulation generates a dynamic dual Voronoi tessellation and cell-centre equations of motion.
Results
The AVM provides a cell-resolution framework for large epithelial systems, naturally handles T1 transitions, and reproduces reported neighbour-number distributions while supporting activity-driven tissue behaviors.
Takeaways & Limitations
The AVM offers a cell-level layer for connecting epithelial mechanics and collective tissue behavior at length and time scales often addressed by continuum models.
Takeaways & Limitations
Activity is introduced through the strong assumption that cells self-propel along their polarity vectors, and extensions to curved or fully three-dimensional surfaces are more involved.
Abstract
from arXiv · showhide
We introduce an Active Vertex Model (AVM) for cell-resolution studies of the mechanics of confluent epithelial tissues consisting of tens of thousands of cells, with a level of detail inaccessible to similar methods. The AVM combines the Vertex Model for confluent epithelial tissues with active matter dynamics. This introduces a natural description of the cell motion and accounts for motion patterns observed on multiple scales. Furthermore, cell contacts are generated dynamically from positions of cell centres. This not only enables efficient numerical implementation, but provides a natural description of the T1 transition events responsible for local tissue rearrangements. The AVM also includes cell alignment, cell-specific mechanical properties, cell growth, division and apoptosis. In addition, the AVM introduces a flexible, dynamically changing boundary of the epithelial sheet allowing for studies of phenomena such as the fingering instability or wound healing. We illustrate these capabilities with a number of case studies.
I. INTRODUCTION
Collective epithelial migration is central to development, wound healing, and disease, yet how cell-level mechanics are coordinated at tissue scale remains unclear. Existing models capture important behaviors but trade off biological realism, cell-level resolution, or computational efficiency; the AVM addresses these limitations by combining active matter with the Vertex Model.
- Motivation: Collective cell migration contributes to embryonic development, wound healing, tumour metastasis, and invasion.
- Motivation: Experiments reveal tissue-scale phenomena including mechanical waves, purse-string contractility, and kenotaxis during epithelial migration and wound closure.
- Open problem: How force generation and transmission are coordinated at tissue level to drive morphogenesis or maintain homeostasis remains unclear.
- Existing models: The Cellular Potts Model supports large systems but uses pixel updates whose dynamics are artificial and difficult to relate to actual cell motion.
- Existing models: The Vertex Model provides cell-resolution descriptions of confluent epithelia, but topology-changing moves require complex mesh restructuring and can produce unphysical repeated flips.
- Contribution: The Active Vertex Model combines the Vertex Model with active matter dynamics, enabling large-system simulations and a natural treatment of T1 transitions.
B. Active Vertex Model
The AVM replaces vertex positions as primary degrees of freedom with cell-centre positions whose Delaunay triangulation defines a dual Voronoi tissue. This preserves local, continuous connectivity changes while making force computation more efficient than repeatedly rebuilding the full Voronoi diagram.
- Motivation: The original Vertex Model is quasi-static, assuming mechanical equilibrium at every instant and therefore not fully describing out-of-equilibrium migration effects.
- Model representation: The AVM tracks cell centres as Voronoi seeds and computes forces from the Vertex Model energy of the resulting cell polygons.
- Efficiency: Computing the full Voronoi diagram at every time step limits the original SPV implementation to several hundred cells, motivating the Delaunay-based alternative.
- Dual representation: The Delaunay triangulation provides a dual Voronoi tessellation, linking cell-centre coordinates to cell vertices and local connectivity.
- Topology changes: Local edge flips retain the Delaunay character during cell-centre motion and naturally implement T1 transitions affecting local cell connectivity.
- Force calculation: The resulting force expression is local and depends only on a cell and its immediate neighbours, allowing standard force-cutoff techniques.
2. Cell alignment
The AVM augments mechanical forces with polarity dynamics, alignment mechanisms, active self-propulsion, stochastic fluctuations, and overdamped cell-centre motion. These ingredients provide alternative ways to model how cell orientation relates to neighbours, migration, or cell shape.
- Polarity: Cell polarity is a unit vector in the xy plane that determines the direction of cell motion and division, distinct from apical-basal polarity.
- Cell alignment: Neighbour alignment minimizes an energy weighted by alignment strength Jp, with the sum taken over each cell’s nearest neighbours.
- Cell alignment: Alternative mechanisms align polarity with migration direction or with the cell’s long axis through the shape-tensor eigenvector.
- Equations of motion: In the overdamped limit, cell-centre motion balances isotropic friction against active driving, mechanical forces, and stochastic forces.
- Equations of motion: The active force fa drives motion along the polarity vector, while stochastic forces represent intracellular and environmental fluctuations.
- Equations of motion: The polarity angle evolves under alignment torque, orientational friction, and orientational randomness, and the equations are solved numerically.
4. Cell growth, division and death
The AVM models epithelial population control through cell growth, division, and death, while rebuilding cell connectivity after demographic events. Its cell-cycle treatment is intentionally simple, and stress-dependent death or extrusion is not included.
- Cell growth: Cell growth is represented as a constant-rate increase in each cell’s native area.The growth rate is η per unit time, applied over simulation step δt.
- Cell death: Cells age by δt each timestep and are removed when they reach a critical age.The current implementation does not make death or extrusion depend on cellular stresses or forces.
- Cell division: Division begins only above critical area Ac, with probability proportional to the excess area A − Ac.After division, daughter cells are placed along the polarity direction, their ages reset, and the Delaunay triangulation is rebuilt.
- Model scope: The model’s cell-cycle description is a simple approximation that could be replaced by more sophisticated extensions.The authors identify later incorporation of richer cell-cycle models as straightforward.
III. EXAMPLES AND APPLICATIONS
The AVM is applied to epithelial-tissue mechanics through T1 transitions, phase behavior, boundary effects, and instabilities. These examples show how activity, tissue parameters, and boundary conditions shape rearrangements and collective states.
- Applications: The AVM validates its use by applying the model to several problems relevant to epithelial tissue-layer mechanics.The applications are intended both to validate the model against related models and to demonstrate its use for biological tissues.
- T1 transitions: T1 transitions occur through Delaunay edge flips, with contacts shrinking to a point before new contacts rapidly expand in the opposite direction.The edge flip occurs when the two relevant pairs of angles each sum to 180°; the illustrated transition occurs in a slowly dynamical, liquid-like regime.
- Activity driven fluidisation phase diagram: Low p0 and low active driving produce an amorphous solid or glassy state in which cells do not exchange neighbours.The phase is characterized using the self-intermediate scattering function and its alpha-relaxation time τα.
- Activity driven fluidisation phase diagram: The solid-like phase begins at p0 = 3.81 at low driving, while open boundaries become liquid-like at much lower active driving than fixed boundaries.For fixed boundaries, the transition extends from p0 ≈3.81 at small fa to a maximum activity; open systems show a maximum transition value fa = 0.03, a factor of 10 compared with the fixed case.
- Activity driven fluidisation phase diagram: Boundary fluctuations reduce rearrangement times, with the open liquid-like system reaching τα ≈10 compared with τα ≈100 for the equivalent fixed system.The authors associate this difference with rearrangements allowed by fluctuating boundaries and expect strong system-size dependence.
- Activity driven fluidisation phase diagram: The AVM produces fingering instabilities in which cell-wide regions migrate outward, while sufficiently high activity can drive eventual cell detachment that the model cannot yet represent.The instability depends on active driving, boundary line tension, noise, and p0; stronger line tension suppresses it, whereas increased noise enhances it.
C. Growth and division
The AVM models epithelial growth and division while tracking the resulting tissue mechanics and neighbour structure. Simulations reveal exponential growth that slows as central pressure builds, alongside neighbour statistics consistent with actual tissues.
- Growth and division: The AVM supports cell division and apoptosis, representing division and death as changes in the Voronoi tessellation.Division adds two cells, while apoptosis or ingression removes one cell.
- Growth and division: The growth simulation ran from 37 to about 24,000 cells over 5000 time units, with cells checked for division every 25 time steps.The simulation used 10^6 time steps of size 0.005 and no active driving.
- Growth and division: Growth was initially exponential but slowed at long times because surrounding cells prevented expansion in the tissue centre.Central cells developed smaller average areas as the tissue expanded.
- Growth and division: Pressure accumulated near the tissue centre, indicating that the later-stage simulated tissue was no longer in mechanical equilibrium.Angular averaging reduced local pressure fluctuations, while substantial central buildup remained.
- Growth and division: The model’s neighbour-count distribution agreed well with observations in actual tissues.The distribution was measured for the tissue after growth.
- Mechanically heterogeneous tissues: Cell-specific mechanical parameters allow heterogeneous tissues, including simulations of red-blue cell sorting driven by junction tensions and random fluctuations.The simulations used 1000 cells, with fixed boundaries and zero active driving.
E. Effects of cellular alignment
The AVM models cellular alignment and shows that different alignment rules produce distinct collective migration patterns across tissue phases and boundary conditions. It also supports non-circular geometries, growth, division, death, and dynamically changing boundaries, while retaining important scope limitations.
- Alignment effects: Velocity alignment produces collective oscillations in confined solid-like tissues and vortex migration in fixed-boundary liquid-like tissues.Without confinement, the same dynamics recover collective polar migration of the cell patch.
- Alignment effects: In an unconfined system, aligning polarity with the cell-shape principal axis also produces collective motion but with significant fluctuations.The fluctuations arise because cell patterns are frustrated by the requirement to remain a Voronoi tessellation.
- Model capabilities: Arbitrary tissue shapes, including annuli and rectangular strips, can model wound healing, void filling, and multiple patches growing toward one another.The annular example is suited to systems where cells migrate to fill a central void.
- Model capabilities: The AVM applies active-matter dynamics to the Vertex Model, enabling large-scale collective migration with cell-centre-based contacts and an efficient implementation.The model includes cell alignment, growth, division, death, and fixed or dynamically changing boundaries.
- Scope and role: The AVM is intended to connect cell-level molecular and mechanical processes with global tissue behaviour while complementing rather than replacing continuum models.Its cell-level resolution is suited to epithelial mechanics at large length and time scales.
- Scope and limitations: The current model omits biochemical signalling, uses a rudimentary self-propulsion assumption, and does not support boundary splitting or merging.These extensions would improve biological scope but can increase computational cost or require biologically plausible general rules.
APPENDIX I: FORCE ON THE CELL CENTRE
The appendix derives the force on a cell centre by differentiating the Vertex Model energy through the Voronoi geometry determined by cell-centre positions. It accounts for direct and neighbour-mediated contributions and adds soft repulsion to regularise liquid-like simulations.
- Force derivation: The cell-centre force is the negative derivative of the Vertex Model energy with respect to the cell-centre position.The derivation handles the fact that the energy is written using Voronoi vertices while the AVM tracks cell centres.
- Force derivation: Moving one cell centre changes its cell and neighbouring cell shapes, so the resulting force cannot be expressed as a simple sum of pairwise interactions.Neighbour-mediated contributions must be included when computing the total force.
- Geometric derivatives: The force calculation uses a Jacobian mapping cell-centre coordinates to dual Voronoi-vertex coordinates, with barycentric coordinates and local geometric derivatives.The appendix derives area, perimeter, and junction-contraction contributions before combining them into the cell-centre force.
- Implementation: The implementation precomputes derivatives for the three particles of each Delaunay triangle and loops over a cell and its immediate neighbours.This reduces computational effort while including only affected Voronoi vertices for neighbour contributions.
- Liquid-like phase: A soft repulsive core prevents cell centres from approaching too closely and thereby regularises simulations in the liquid-like phase.The repulsion prevents unphysical triangulation self-intersections and does not otherwise interfere with AVM dynamics.
APPENDIX II: ALGORITHM FOR HANDLING BOUNDARIES
The boundary algorithm dynamically expands or contracts the epithelial sheet as internal cell-centre motion changes the required boundary. Local geometric checks trigger particle insertion or removal while avoiding discontinuous changes in cell shape and centre forces.
- Boundary handling: Because internal dynamics change the required boundary, the AVM adds and removes boundary particles instead of retaining a constant boundary-particle count.This permits the boundary line to contract or extend as the tissue changes.
- Boundary expansion: Boundary expansion is triggered when the angle opposite a boundary edge exceeds 90°, indicating that a dual Voronoi vertex would spill outside the boundary.A mirrored particle is added, the edge is removed, and the new particle is connected to the surrounding boundary particles before retriangulation.
- Boundary handling: The boundary-particle addition and removal procedures converge in a single step.The stated procedure is designed to update the boundary locally as the tissue configuration changes.
- Boundary shrinking: Boundary shrinking removes particles that have no connections to internal particles or have at most two incident edges.Such particles can be safely removed because no part of a cell occupies the associated boundary triangle.
- Boundary limitations: Boundary updates avoid sudden discontinuous changes in cell shape or the force on cell centres, although forces on boundary particles can change discontinuously.Smoothly turning on or fading out interactions is proposed as a possible mitigation.
APPENDIX III: IMPLEMENTATION
The AVM is implemented in the SAMoS codebase, whose organisation and AVM implementation are documented in this appendix.
- Implementation: The AVM is implemented into the SAMoS code developed by the authors.The appendix provides a general overview of SAMoS organisation and discusses how the AVM is implemented within it.
SAMoS overview
SAMoS is a modular, object-oriented C++ package for agent-based active-matter simulations, with components governing system setup, interactions, alignment, integration, populations, constraints, and output. AVM simulations use configuration, data, and boundary inputs, while the current implementation runs on one CPU core.
- Architecture: SAMoS is a modular, object-oriented C++ package for simulating agent-based active matter on flat or curved surfaces.Its loosely coupled components are designed to support testing, debugging, and extensions.
- Core components: The System, Parser, Messenger, Neighbour list, Constraint, Interactions, and Integrator components manage configuration, execution, messaging, neighbours, surfaces, forces, and motion.Components are implemented as class hierarchies, with interactions supporting multiple simultaneous and type-specific interaction types.
- Simulation controls: Alignment, Population, Dump, and Log components handle orientation, particle addition or removal, output formats, and system-state reporting.Population controls can model cell division and death, while multiple dump types or population controls may operate simultaneously.
- Inputs: SAMoS simulations require a data file and configuration file, while AVM simulations additionally require a boundary file defining initial boundary labels and connectivity.Configuration commands are parsed and executed in their listed order.
- Execution: The current SAMoS implementation runs on a single CPU core.
AVM implementation
The AVM extends SAMoS with a Mesh class that maintains the cell-centre triangulation, derives its dual Voronoi diagram, and supplies geometric information for force calculations. Constrained Delaunay triangulation incorporates the user-specified tissue boundary, while feedback keeps mesh geometry synchronized with cell-centre positions.
- AVM implementation: The AVM adds a lightweight half-edge Mesh class that stores the Delaunay triangulation and computes its dual Voronoi diagram.The Mesh class also preserves the Delaunay character between rebuilds through equiangulation.
- AVM implementation: A feedback loop from the integrator keeps the Mesh class synchronized with the current positions of cell centres.
- AVM implementation: The initial cell-centre mesh uses CGAL’s constrained Delaunay triangulation, with the user-supplied boundary line acting as a triangulation constraint.The implementation accounts for the cell-sheet boundary while constructing the initial triangulation.
- AVM implementation: The force calculation obtains dual-vertex positions and Jacobian components from the Mesh class to compute forces on cell centres.
- AVM implementation: Figure 14 summarizes the general organization of the AVM implementation within SAMoS.