A geometric framework for curvature-dependent collective behavior of polar active agents on curved surfaces

TL;DR

This work addresses curvature-driven changes in collective motion by introducing a geometry-based model for polar active agents on curved surfaces. It builds a Vicsek-like alignment framework using intrinsic surface geometry and parallel transport to compare neighbor velocities, and analyzes both single-agent dynamics on spheres and spheroids as well as many-agent swarms with alignment interactions. Key findings show disorder–order transitions on both spheres and spheroids, with the transition point rising with curvature heterogeneity; for highly non-spherical surfaces swarms localize near the equator and exhibit orientation fluctuations dependent on curvature. The approach yields a minimal, extensible tool for interpreting curvature-influenced collective behavior in biological systems, with potential applications to cell sheets and cytoskeletal organization on curved membranes or tissues, and can be extended to deformable surfaces. $K$ and $ obreaks \\sigma$ govern the transition, while $oldsymbol{P}$ tracks global polar order and $V_{ extrm{sph}}$ monitors orientational stability.$

Abstract

In biological systems, active agents such as actomyosin and cells move and interact on curved surfaces, exhibiting diverse phenomena. These observations have motivated studies of how curvature shapes their collective behavior. Here, using a geometric framework, a minimal model is presented for interacting active agents on curved surfaces with Vicsek-like polar alignment. A transition between disordered and ordered states occurs on spheres as well as on oblate and prolate spheroids. As the deviation from sphericity increases, the transition point shifts to higher alignment strengths, and swarming localizes to an equatorial belt away from the poles, indicating that curvature heterogeneity influences the emergence of the polar-ordered state.

Figures (4)

• Figure 1: (a) A trajectory of an active agent on a sphere of radius $R=10$ for $t=5000$. $\sigma=1$, $v_0=1$, and $I=100$. (b) The probability distributions of the azimuthal angle $\varphi$ (top) and $Z$ (bottom). $\sigma=1$, $v_0=1$, and $I=100$. (c) The mean squared displacement of active agent on sphere of radius $R=\sqrt{0.1}, 1, \sqrt{10}, 10, \sqrt{1000}$, or $100$. The theory line is given by $4 \left(t + 2 \left(\exp{(-{t}/{2})} - 1\right)\right)$.
• Figure 2: (a,b) Active agent on oblate and prolate spheroids of $A=10, C=3$ and $A=3, C=10$, respectively, for $t=3000$. $\sigma=1, v_0=1, I=100$. (c,d) The probability distributions of the azimuthal angle $\varphi$ (top) and $Z$ (bottom) for oblate and prolate spheroids, respectively. The theoretical lines given by Eq.(\ref{['eq:distribtionOfZ']}) are indicated by red lines.
• Figure 3: (a,b,c) Snapshot images of polar interacting active agents on sphere of radius $R=10$ for three different time points $t_0$ (red), $t_0+3$ (green) and $t_0+6$ (blue). Arrows indicate the velocity vectors. $N=1000$, $\sigma=0.2$ and $I=10$. The interaction strength are given by (a) $K=0.05$, (b) $0.1$ and (c) $1.0$. (d) Polar order parameter $\lVert\mathbf{P}\rVert$ plotted against $K$ for different value of $\sigma$. (e) The phase diagram in $K-\sigma$ plane showing the transition from disorder to order phases. The polar order parameter $\lVert\mathbf{P}\rVert$ is color encoded.
• Figure 4: (a,b,c,d) Snapshot images of polar interacting active agents on oblate (a,b) and prolate (c,d) spheroids for three different time points $t_0$ (red), $t_0+3$ (green) and $t_0+6$ (blue). Arrows indicate the velocity vectors. $N=1000$, $K=0.2$, $\sigma=0.2$, $R=10$, and $I=10$. The eccentricity $e$ is given by (a,c) $e=0.5$ and (b,d) $e=0.9$. (e) Polar order parameter $\lVert\mathbf{P}\rVert$ plotted against $K$ for oblate and prolate spheroids with different value of eccentricity $e$. $\sigma=0.2$, $N=1000$. (f) The $K$-values that give $\lVert\mathbf{P}\rVert=0.2, 0.3, 0.4$ are plotted against eccentricity $e$. Interpolation was used to obtain the $K$-values. (g) The distribution of $\hat{P}_Z$ for different value of eccentricity $e$ for oblate and prolate spheroids. $K=0.1$, $\sigma=0.1$, $N=1000$, $R=10$ and $I=10$. The distributions are obtained from 100 samples. (h) The spherical variance of the temporal fluctuation in the unit polar order vector $\hat{\mathbf{P}}$ averaged over 100 samples plotted against eccentricity $e$ for oblate and prolate spheroids. $K=0.1$, $\sigma=0.1$, $N=1000$, $R=10$ and $I=10$.