Extended Data Fig. 5: Emergent global motion modes in modelled active solid under isotropic lateral confinement.

(a) Schematic diagram of the bead–spring model for active solid under two-dimensional isotropic lateral confinement (Methods). The model consists of N = 511 self-propelled particles (black solid circles). Every nearest-neighbour pair of particles is connected by an interparticle spring with spring constant k_b (red). The particles also experience elastic forces due to substrate adhesion and lateral spatial confinement (see main text) via a restoring spring (green) and a boundary spring (blue) with spring constant k_s and k_r, respectively. The three spring constants together determine the system’s local elasticity. In simulations the interparticle spring constant k_b was used as a proxy for the system’s local elasticity, with the ratios between k_b, k_s and k_r fixed. (b,c) Representative trajectories of particles in the modelled active solid that underwent global oscillatory translation (panel b) and oscillatory rotation (panel c). Most particles (except those very near the centre or the boundary) followed periodically oscillating quasi-circular trajectories (at relatively high activity; panel b; Supplementary Video 9) or quasi-linear concentric trajectories (at relatively low activity; panel c; Supplementary Video 10) with highly synchronized phases (insets of panel b,c), in the same manner as the motion of matrix-embedded cells in the experiments undergoing global oscillatory translation (main text Fig. 1a) or rotation (main text Fig. 1b), respectively. Black dot in each panel indicates the centre of the simulation domain. Scale bars represent 1/3 of the interparticle distance at equilibrium and colour map indicates time. Insets: Oscillation phases of individual particle’s velocity components plotted in the same way as in insets of Fig. 1a,b. Simulation parameters:v₀ = 15 (panel b) or v₀ = 3 (panel c), k_b = 12,k_r = 0.6, and k_s = 0.044. (d,e) Temporal dynamics of spatially averaged particle velocity in the modelled active solid in global oscillatory translation mode (panel d) or oscillatory rotation mode (panel e). The velocity was averaged over all particles in the simulation and then decomposed as Cartesian (yellow and blue traces) and polar-coordinate components (red: tangential or azimuthal component; green: radial component). The spatially averaged particle velocity in the two emergent modes was characterized by distinct temporal dynamics in Cartesian or polar coordinates similar to that found in the experiments (main text Fig. 1e,f). Simulation parameters are identical to those used in panels b,c.