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From flocking to jamming in collective cell dynamics: a Vicsek-like model including contact forces

Laurent Navoret, Roxana Sublet, Marcela Szopos

TL;DR

This work develops a Vicsek-like agent-based model that incorporates hard-contact constraints and soft attraction-repulsion to capture the transition between flocking and jamming in collective cell dynamics. The authors formulate the dynamics for positions, velocities, and polarities, prove well-posedness for a regularized version using differential inclusions, and implement a three-stage semi-implicit discretization with a Uzawa-projection step to enforce non-overlap. A comprehensive numerical study demonstrates the model’s ability to reproduce order–disorder transitions, boundary-driven rotation, and density- and geometry-dependent jamming, as well as how obstacles and elastic interactions modulate these regimes. Overall, the framework provides a mathematically grounded and computationally tractable tool for understanding tissue-scale flows and emergent cell dynamics under confinement and interaction constraints.

Abstract

The goal of the present work is to propose an agent-based model that originally combines classical Vicsek-like polarity alignments and contact forces, as implemented in the framework developed by Maury and Venel in [Maury, Venel, 2011]. The description additionally incorporates velocity feedback on polarity and soft attraction-repulsion interactions. After carefully studying the well posedness of the model, we introduce a suitable discretization and perform an extensive range of numerical experiments to assess the impact of different modeling ingredients. The dynamical system is capable of recovering the order-disorder phase transition of the flock, as well as the jamming effect in high density regimes. As such, the developed framework can be seen as a promising theoretical tool that could contribute to improving the understanding of complex collective cell dynamics and emerging tissue flows.

From flocking to jamming in collective cell dynamics: a Vicsek-like model including contact forces

TL;DR

This work develops a Vicsek-like agent-based model that incorporates hard-contact constraints and soft attraction-repulsion to capture the transition between flocking and jamming in collective cell dynamics. The authors formulate the dynamics for positions, velocities, and polarities, prove well-posedness for a regularized version using differential inclusions, and implement a three-stage semi-implicit discretization with a Uzawa-projection step to enforce non-overlap. A comprehensive numerical study demonstrates the model’s ability to reproduce order–disorder transitions, boundary-driven rotation, and density- and geometry-dependent jamming, as well as how obstacles and elastic interactions modulate these regimes. Overall, the framework provides a mathematically grounded and computationally tractable tool for understanding tissue-scale flows and emergent cell dynamics under confinement and interaction constraints.

Abstract

The goal of the present work is to propose an agent-based model that originally combines classical Vicsek-like polarity alignments and contact forces, as implemented in the framework developed by Maury and Venel in [Maury, Venel, 2011]. The description additionally incorporates velocity feedback on polarity and soft attraction-repulsion interactions. After carefully studying the well posedness of the model, we introduce a suitable discretization and perform an extensive range of numerical experiments to assess the impact of different modeling ingredients. The dynamical system is capable of recovering the order-disorder phase transition of the flock, as well as the jamming effect in high density regimes. As such, the developed framework can be seen as a promising theoretical tool that could contribute to improving the understanding of complex collective cell dynamics and emerging tissue flows.

Paper Structure

This paper contains 21 sections, 1 theorem, 34 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Description and analysis of the model
  3. Positions and velocities dynamics.
  4. Polarities dynamics.
  5. Well-posedness.
  6. Numerical method
  7. Discretization of the positions and velocities dynamics.
  8. Discretization of the polarities dynamics.
  9. Numerical experiments and discussion
  10. Order-disorder phase transition in a periodic domain
  11. Rotation dynamics in a disk
  12. Influence of the domain shape and jamming effects
  13. Influence of obstacles.
  14. Influence of the soft attraction-repulsion force
  15. Conclusion and outlook
  16. ...and 6 more sections

Key Result

Proposition 1

Let $Q = \{ \mathbf{X} \in \mathbb{R}^{2N} \mid \forall i < j, \, D_{ij}(\mathbf{X}) \geqslant 0 \}$ be the set of admissible configurations and assume that $\mathbf{U}$ is Lipschitz and bounded. Then, for any initial data $(\mathbf{X}_0, \theta_0) \in Q \times \mathbb{R}^N$ and any time $T > 0$, th Moreover, this is also the unique absolutely continuous solution to Eq. eq:proj_complet.

Figures (11)

  • Figure 1: Schematic presenting the geometrical configuration. $(\mathbf{X}_i, \mathbf{V}_i, \mathbf{P}_i)$: position, velocity and polarity of the $i$-th cell. $D_{ij}(\mathbf{X})$: distance between the $i$-th and $j$-th cells. $D_b(\mathbf{X}_i)$: distance between the $i$-th cell and the boundary.
  • Figure 2: (Periodic boundary condition) Order parameter as a function of the diffusion for two different density values, $\rho = 0.707$ (left panel) and $\rho = 0.839$ (right panel). Mean (solid lines) and interquartile interval over $20$ simulations with $T = 20\,h$ and different time steps. The vertical line at $D =0.96\ \text{rad}^2 \,\text{h}^{-1}$ corresponds to the reference diffusion coefficient used in the other test-cases.
  • Figure 3: (Disk domain) Cell configuration at final time $T=20\,\text{h}$, with different values of $\delta$ and $\mu$. Parameters: $\Delta t = 10^{-2}\,\text{h}$, $N= 160$, cell density $\rho = 0.707$. (Left panels) Polarities are represented by red vectors, velocities by orange vectors and the contact forces by edges between the cells where they are activated, colored according to the magnitude of the contact force. (Right panels) Normalized global mean speed (in red), rotation order parameter (in blue) and order parameter (in green) as functions of time. See also the corresponding Suppl. Mov. 1, Suppl. Mov. 2 and Suppl. Mov. 3.
  • Figure 4: (Disk domain) Mean speed, order parameter and rotation order parameter as functions of parameters $\delta$ and $\mu$. Each value is the average over $20$ simulations with parameters $\Delta t = 10^{-2} \text{h}$, $N = 160$, cell density $\rho = 0.707$.
  • Figure 5: (Disk domain) Rotation order parameter as function of time in different regions of the disk (depicted in the left panel), for two different values of the parameter $\mu$: $\mu = 0\ \text{rad}\, \text{h}^{-1}$ (middle) versus $\mu = 6.2\ \text{rad}\, \text{h}^{-1}$ (right). Parameters: $\Delta t = 10^{-2}\,\text{h}$, $N= 160$, cell density $\rho = 0.707$.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2: Convergence analysis of the Uzawa algorithm