Boundaries Program Deformation in Isolated Active Networks

TL;DR

This paper demonstrates that boundary geometry actively controls deformation in isolated active MT-kinesin networks, challenging the view of boundaries as passive constraints. A coarse-grained hydrodynamic-like model couples boundary shape to internal stresses via mass conservation, predicting both shape-preserving and shape-changing contractions and explaining the universality across geometries. Experiments show self-similar contraction across diverse boundary shapes and show programmable deformation via spatial and temporal modulation of optical activity patterns. These results establish boundary geometry as a powerful design parameter for programmable deformation in synthetic active matter and offer insights into boundary-driven organization in biological systems.

Abstract

Cellular structures must organize themselves within strict physical constraints, operating with finite resources and well-defined boundaries. Classical systems demonstrate only passive responses to boundaries, from surface energy minimization in soap films to strain distributions in elastic networks. Active matter fundamentally alters this paradigm - internally generated stresses create a bidirectional coupling between boundary geometry and mass conservation that enables dynamic control over network organization. Here we demonstrate boundary geometry actively directs network deformation in reconstituted microtubule-kinesin systems, revealing a programmable regime of shape transformation through controlled boundary manipulation. A coarse-grained theoretical framework reveals how boundary geometry couples to internal stress fields via mass conservation, producing distinct dynamical modes that enable engineered deformations. The emergence of shape-preserving and shape-changing regimes, predicted by theory and confirmed through experiments, establishes boundary geometry as a fundamental control parameter for active materials. The control principle based on boundaries advances both the understanding of biological organization and enables design of synthetic active matter devices with programmable deformation.

Figures (4)

• Figure 1: Optical-control protocol first activates cross-linking motor proteins to form the MT networks and isolates the network from embedding solution, allowing them to contract. (a) shows the active matter system used in this study, which consists of fluorescently labeled, stabilized MT filaments and kinesin motors that cross-link under illumination. An initial pulse of light activates motor proteins within a region of illumination. Activated motor proteins crosslink the MTs and form a contractile network. Isolation of the network from the solutions requires a second pulse at around $\sim 50-80$s. (b) shows the macroscopic (top row) and microscopic (second row) snapshots of the network, from left to right: during the activation and network formation, at the time of isolation, and shape preserving contraction. The colored dots in the second row track the loci of four distinct microscopic asters in time. Scale bar for top row, $500 \mu$m; Scale bar for second row, $50 \mu$m. (c) shows the profile of the contracting network in time (horizontal axis). The major three phases of the dynamics are separated by white vertical lines. (d) For six different boundary geometries the contraction of networks is portrayed by overlaying the networks’ boundaries as they shrink in time. Scale bar, $500 \mu$m. (e) The relative area of networks in (d) decays over time. The gray dashed curve is an exponential fit to the data, with $\bar{\chi} = 0.89 \pm 0.01, \bar{\tau} = (41 \pm 2)$s.
• Figure 2: Comparison and agreement between experiments and theory supports the role of activity-induced viscous interaction in mass-conserved contraction. (a) For three geometries of hexagon, hexagram, and ellipse, the flow fields depicted with streamlines are shown as extracted via PIV in experiments (left) and simulated (right). The velocity fields are approximately linear and radial in all cases. Scale bar, 100$\mu$m. (b) shows the relation between the radial velocity $v_r$ with the distance from the mass center $r$. The velocity fields are non-linear in phase II (circles), but become strongly linear in phase III (squares). (c) shows the cross-correlations between the simulated velocity fields $\vb*{v}_{sim}$, and the experimental velocity fields by PIV analyses $\vb*{v}_{piv}$ over the course of the network’s contraction. The cross-correlation, defined in (c), where $\langle f \rangle = \int\dd^2x\,f(\vb*{x})$, is normalized, hence bounded between $[-1, +1]$. The correlations decrease due to the decreasing network areas and fixed precision of PIV analyses (Supplementary Information). (d) The radial velocity distribution of an ellipse network at $100s$ in (a). The radial velocity is normalized as $\tilde{v}_r = (2v_r - \max{v_r} - \min{v_r}) / (\max{v_r} - \min{v_r})$, and the position is normalized as $\tilde{r}_x = r_x / a, \tilde{r}_y = r_y / b$, where $a, b$ are the lengths of semi-major axis and semi-minor axis, hence both the normalized radial velocity and position are bounded between $[-1, +1]$. The black dashed line shows the simulation result, which follows a power-law scaling of $-1$. The shaded regions (outer along the major axis and inner along the minor axis) delineate the near-boundary zones where the experimental data deviate from the simulation due to the boundary mechanics.
• Figure 3: Boundary mechanics provides forces, guiding the deformation of the active network. (a) shows the contractile behavior of a MT network in L-shape. Scale bar, 200$\mu$m. (b) shows the measurement of the tracked angle $\theta_L$. Two white straight lines represent the fit lines of two edges by the regression methods. Scale bar, 200$\mu$m. (c) shows the tracked angle $\theta_L$ changes with time. Linear regression (green) and random sampling and consensus (RANSAC) regression (purple) are used to measure the angle respectively (Supplementary Information). The data is well fitted by a quadratic function $\theta_L = (-0.0019 \pm 0.0001)t^2 + (0.73 \pm 0.03)t + (57.5 \pm 2.2)$ in degrees (gray dashed line). The corresponding angular velocity $\omega_L = (-0.0038 \pm 0.0002)t + (0.73 \pm 0.03)$ (inset). (d) shows the schematic and experimental snapshot of MT network boundary. The outermost MTs are arranged with a strong perpendicular orientation to the boundary, performing the passive response of the network. Scale bar, 10$\mu$m.
• Figure 4: Contraction of the network can be programmed via modulating the pattern of illumination in space and time. (a) shows purely-spatial modulations of light, where the top segment of the rectangle is illuminated more/less strongly. Greater intensity of light activates larger number of motor proteins and thus generates larger active stresses, which leads to larger and faster contraction on the brighter side, and causes bending. Scale bar, 200$\mu$m. (b) The pattern of illumination varies in time to interpolate between the two static patterns of (a). Using this dynamic modulation we manage to change the bending direction as the network contracts. Scale bar, 200$\mu$m.