Flow-wave coupling synchronizes oscillations in growing active matter

TL;DR

The results show that mechanical forces actively maintain the coherence of biochemical waves, providing a general mechanism for long-range order in oscillating active matter.

Abstract

Oscillatory biochemical signals and mechanical forces must coordinate robustly during development, yet the principles governing their mutual coupling remain poorly understood. In syncytial embryos and cell-free extracts, mitotic waves propagate across millimeter scales while simultaneously generating cytoplasmic flows, suggesting a two-way interaction between chemical oscillators and mechanics. Here, we combine experiments in Xenopus Laevis cytoplasmic extracts with a minimal particle-based model to reveal a mechanochemical feedback that stabilizes phase wave propagation. In contrast to previous models of oscillatory active matter, an asymmetric size cycle, slow growth and rapid shrinkage, combined with size-dependent mechanical interactions generates a net particle displacement and flows aligned with the wave direction, which in turn drive a synchronization transition. Our results show that mechanical forces actively maintain the coherence of biochemical waves, providing a general mechanism for long-range order in oscillating active matter.

Figures (3)

• Figure 1: Centrosome flows driven by microtubule polymerization are phase-locked to the mitotic wave in cytoplasmic extract. (a) Cropped microscopy images showing asters that grow, shrink, and mechanically interact. Scale $200\mu m$. (b) Tubulin-intensity image of the extract, revealing alternating polymerized (bright) and depolymerized (dark) regions. A dotted orange iso-phase line marks the mitotic wave, and the arrow shows the direction of wave propagation. The inset shows bead trajectories (green) indicating correlated flows. (c) Phase field extracted from microtubule intensity experiment, together with corresponding velocity fields extracted from the tracking of the beads. Dashed lines mark constant-phase contours. (d) Mean particle speed as a function of phase, obtained by spatiotemporal averaging. Speeds peak near the end of the cycle. (e) Probability density of the angle between the phase gradient (opposite to the wave direction) and velocity (average over five experiments, measured over 1--3 cell cycle) at $0$ and $\pi$, demonstrating that flows align parallel or antiparallel to the wave (f) Schematic of the agent-based model: two-dimensional particles grow at rate $v_p$ and depolymerize instantaneously when their internal phase $\varphi_i$ reaches $2\pi$, while interacting via a soft repulsive potential $U$ (g) Simulation snapshot of $N=2500$ particles exhibiting self-organized dense and dilute regions; inset: particle tracks. ( $T_\mathrm{osc}/T_\mathrm{grow}$$=1.2$, $\omega=1$, $\sigma=0.2$, $\epsilon=0.5$, $D_\varphi/\epsilon=0.01$, $\mu U_0=2$, $r_0/\ell_0=0.05$) (h) Phase and velocity fields extracted from simulations, revealing that absolute velocity is maximal at the end of the cycle in simulation (i) and the flows align with the wave direction (j).
• Figure 2: Particles flow along the phase gradient. (a) One-dimensional chain with periodic boundaries. A phase wave of wavelength $L$ induces periodic growth and shrinkage of the particles. When particles are in contact, they form a dense cluster whose front moves in the direction of the wave and whose rear moves in the opposite direction. Colors indicate instantaneous velocity relative to the wave direction (blue: trigonometric; red: opposite). The circled particle shows a net forward drift over one cycle. (b) Mean displacement along the phase gradient, $\langle x_\parallel(t)\rangle$ (Eq. \ref{['eq:x_parallel_definition']}) in the comoving frame. Particles exhibit oscillatory motion with a net drift $\Delta_\parallel$ per cycle. The dashed green curve shows the 1D system without size-dependent mobility. (c) 2D system with periodic boundaries. A planar wave is imposed along one box direction. Left: particle phases (same color map as Fig. \ref{['fig:xp_model']}); right: arrows show the particle velocity, and the color map show the values of velocity projected along the wave direction. A dense band forms where particles are large and in contact, and flows are localized in this band, leading to oscillatory motion shown in (d). (e) 2D circular system with a central phase defect. The phase winds around the defect, generating a tangential phase gradient. A dense wedge forms and particles flow along this tangential gradient, leading to oscillatory motion shown in (f). (g,h) Net drift $\Delta_\parallel$ (Eq. \ref{['eq:delta_parallel_definition']}) a function of the growth number $T_\mathrm{osc}/T_\mathrm{grow}$ or the wavelength $\lambda$. Drift occurs only when particles grow sufficiently during one oscillation period to come into contact with their neighbors, and decreases with the wavelength.
• Figure 3: Mechanical–oscillatory coupling maintains order in oscillating active matter. (a) Steady-state phase fields for varying synchronization strength $\epsilon$ and growth number $T_\mathrm{osc}/T_\mathrm{grow}$. The correlation length $\xi$ depends strongly on both parameters. The other parameters are indicated in Fig. \ref{['fig:xp_model']}. $t_\mathrm{simu}/T_\mathrm{osc}=120$. (b) Phase diagram of $\xi$ (color). Spatial coherence increases with synchronization until full phase alignment is reached, after which $\xi$ decreases again. (c) Phase diagram of the neighbor-exchange timescale $\tau$ (color). At high $T_\mathrm{osc}/T_\mathrm{grow}$, particles exchange neighbors rapidly, indicating substantial mixing. (d) Kuramoto parameter $\langle k \rangle$ (Eq. \ref{['eq:kop_definition']}) a function of the growth number, revealing a mechanically induced synchronization transition. (e) Rescaling the Kuramoto parameters with the synchronization strength collapses the curve (f) The average displacement after one period $\Delta/\ell_0$ (Eq. \ref{['eq:delta_definition']}) with synchronization strength, indicating that synchronization suppresses diffusive motion. This is consistent with observations of the previous section: increasing $\epsilon$ increases the typical length scale, and this decreases the net displacement. (g) Rescaling $\Delta$ by its value in the asynchronzied regime reveals a chemically induced slow down transition.