Collective spatial reorganization from arrest to peeling and migration through density-dependent mobility in internal-state coordinates

Abstract: Numerous problems in development, regeneration, and disease involve simultaneous evolution of both spatial organization and the internal state of the constituents in addition to local interactions and crowding. This motivates us to study a minimal model for interacting populations evolving in coupled spatial and internal-state coordinates. We focus on a specific transition of particular biological interest: the reorganization of dense collectives from compact or arrested states toward boundary-led peeling and migration. In our formulation, each particle carries a spatial position and a scalar internal state, and interacts through finite-range forces. Mobilities are defined on both spatial and internal-state coordinates with a density dependence, and are asymmetrically cross-coupled. We derive update equations for stochastic dynamics in the overdamped limit and perform numerical simulations. We find that mobility in internal-state coordinates alone provides an independent control axis for large-scale spatial reorganization. In particular, increasing the baseline internal-state diffusivity and tuning its density dependence drives a transition from arrested aggregates to a peeling-like regime with broad spatial excursions, strong outward radial bias, and edge-localized activity, while the baseline positional diffusivity is held fixed. The transition is accompanied by correlated broadening of spatial and internal-state displacements, systematic reorganization of radial density and density-curvature profiles, and a pronounced dependence on system size, consistent with the idea that growing aggregates can cross into a boundary-dominated migratory state. These results establish the utility of our approach and motivate a broader framework aimed at modeling state change, spatial redistribution, and neighborhood structure within a common formalism.

• The paper shows that increasing internal-state diffusivity induces a sharp switch from static aggregates to peeling, migratory behavior while spatial diffusivity remains fixed.
• The study employs a stochastic particle model with asymmetric density-dependent mobility, revealing strong correlations between spatial and internal-state excursions.
• Key findings highlight boundary-localized activity and system size effects, offering insights into tissue morphogenesis and collective cell migration.

Collective Spatial Reorganization via Density-Dependent Internal-State Mobility

Overview and Motivation

This paper introduces and analyzes a stochastic particle model coupling real-space positions and internal states, where density-dependent mobilities act asymmetrically between these coordinates. The motivation is to capture essential features of biological collectives, such as tissues or organoids, in which not only spatial organization but also cell-intrinsic states co-evolve, yielding phenomena including bulk-to-boundary transitions, peeling, and migration, that cannot be explained by spatial interactions alone.

The central technical focus is to determine whether mobility in internal-state coordinates alone can trigger a qualitative collective reorganization from arrested aggregates toward boundary-driven migratory regimes, all while the baseline spatial diffusivity remains fixed.

Model Formulation

The model considers $N$ particles, each with a position $\mathbf{r}_i \in \mathbb{R}^2$ and a scalar internal coordinate $\tau_i$. Interactions are defined via a pairwise Hamiltonian $U(\mathbf{r}_i - \mathbf{r}_j, \tau_i, \tau_j)$ with a Hertzian hard core, a smooth attractive shell modulated by similarity in $\tau$, and a quartic double-well intrinsic potential in $\tau$. Density dependence is incorporated by letting both positional and internal-state mobilities decrease as the local density increases, reflecting reduced rearrangement in crowded environments.

Crucially, cross-coupling between spatial and internal-state degrees of freedom is introduced in the equations of motion. In the regime studied, only an asymmetric deterministic cross term is present: positional dynamics is influenced via a local density-curvature signal projected on the particle's conservative drift, whereas internal-state dynamics is not directly coupled to positional fluctuations.

Stochastic terms are sampled from a local Gaussian with a covariance derived from the mobility matrix, ensuring consistency with the (possibly asymmetric, nonreciprocal) dissipative structure.

Key Results and Numerical Findings

1. Internal-State Mobility Governs Collective Spatial Reorganization

By varying only the baseline diffusivity in internal state $D_\tau^0$ (with baseline positional diffusivity $D_r^0$ held fixed), the system exhibits a sharp transition from spatially arrested aggregates to peeling-like, outwardly migrating regimes. Quantitatively, the late-time mean spatial displacement (MSD) increases rapidly with $D_\tau^0$, whereas $\beta_\tau$ (controlling density-dependence strength) modulates properties secondarily except near the crossover.

2. Correlation Between Spatial and Internal-State Excursions

Joint distributions reveal that in the high-mobility regime, particles display strongly correlated broadening of both spatial and internal-state displacements. The onset of large spatial excursions coincides with transitions in $\mathbf{r}_i \in \mathbb{R}^2$0, evidencing mutual reinforcement between internal-state transitions and collective migration.

3. Local Structure and Edge-Bias in Activity

Radial profiles demonstrate that aggregate cores remain largely static, while collective reorganization is initiated and sustained predominantly at the boundary, as captured by strong spatial excursions and local broadening of $\mathbf{r}_i \in \mathbb{R}^2$1 near the rim. This is quantified via density-curvature proxy fields and the distribution of signed radial displacements, which become strongly outwardly biased after the transition.

4. System Size and Interaction Sector Dependence

Increasing system size $\mathbf{r}_i \in \mathbb{R}^2$2 intensifies peeling-like behavior in the migratory regime, highlighting a size threshold for onset of boundary-dominated motion. In interaction parameter sectors promoting attraction between like internal states, aggregates remain compact and less diffusive, while in more repulsive or mixed regimes, transitions to migratory behavior occur at smaller $\mathbf{r}_i \in \mathbb{R}^2$3.

5. Robustness to Cross-Coupling Strength

Once the system has crossed into the migratory phase, further modifications to the cross-term scaling $\mathbf{r}_i \in \mathbb{R}^2$4 produce only moderate quantitative effects; the primary control is the internal-state diffusivity.

Theoretical and Practical Implications

This work demonstrates that internal-state mobility acts as an independent control axis for large-scale collective reorganization, even in the absence of direct changes to spatial transport properties. The boundary-localized onset of collective migration and its correlation with internal-state transitions provide a plausible minimal framework for understanding tissue or organoid morphogenesis, particularly transitions involving escape, peeling, or migration localized at edges—consistent with observations in development, regeneration, and disease contexts.

On a theoretical level, the asymmetric, density-curvature–driven coupling exemplifies how nonreciprocal, nonequilibrium dynamics generalize Onsager-structured models in biology and condensed matter, producing behaviors not possible in reciprocal, purely passive systems.

Future Directions

Several avenues emerge:

• Incorporating Alignment/Polarity: The model could be extended with vectorial internal degrees of freedom to investigate collective migration with alignment or flocking, relevant for active matter and morphogenetic transitions.
• Memory and Irreversibility: Introducing path-dependent or irreversible transitions in $\mathbf{r}_i \in \mathbb{R}^2$5 would model irreversible differentiation or fate commitment.
• Data-Driven Parameterization: The framework is directly amenable to coupling with modern high-content datasets from spatial transcriptomics and live imaging, potentially allowing for inference of effective mobility and interaction fields from empirically observed cell trajectories.
• Geometric and Mechanical Feedback: Coupling spatial organization with explicit mechanical fields or substrate geometry would expand applicability to morphogenesis under stress, growth, or curvature constraints.

Conclusion

The analysis establishes internal-state mobility—modulated by local density—as a decisive regulator of group-scale physical reorganization in coupled dynamical systems. The findings reframe spatial evolution and state transitions not as sequential or separable, but as components of a tightly coupled feedback process. This perspective provides a minimalist yet powerful modelling platform for the broad class of biological and synthetic systems where identity progression and spatial redistribution are inextricable.