Spatial patterning of force centers controls folding pathways of active elastic networks

TL;DR

Problem: how the spatial distribution of active force centers affects folding pathways and mechanical stability in active elastic networks. Approach: Langevin dynamics simulations of a triangular bead-spring network with patterned activity (active core vs active periphery) and comparison to a uniform distribution, using observables such as the radius of gyration $R_g$, stress, and irreversibility metrics. Key findings: patterned activity prevents full collapse at large forces, with $R_g$ exhibiting high-force scaling $R_g \sim (1-\phi)^\nu$ that differs by pattern ($\nu=2.5$ for active core, $\nu=1.75$ for active periphery); irreversibility thresholds depend on distribution and are tied to crease formation; folding trajectories are robust to stochastic timing but highly sensitive to defects. Significance: demonstrates that localization or delocalization of active stresses provides a tunable mechanism to control rigidity and folding in elastic networks, with implications for biological morphogenesis and the design of programmable metamaterials; future work includes 3D extensions and adding bending/excluded-volume interactions.

Abstract

We study the effect of the spatial distribution of active force dipoles on the folding pathways and mechanical stability of rigid-elastic networks using Langevin dynamics simulations. While it has been shown in Majumdar et al., J. Chem. Phys. 163, 114902 (2025) that a sharp collapse transition is evident in triangular (elastic) bead-spring networks under the action of contractile (or extensile) force dipoles distributed randomly across the network, here, we show that when the spatial distribution is correlated, e.g., like a patch in the center (``active core'' model) or a band-like distribution along the periphery (``active periphery'' model), the network undergoes only a partial decrease in size even at large forces, thereby showing an enhanced mechanical stability just from a spatial rearrangement of the active dipoles. Further, an active periphery network shows higher mechanical stability initially, for a range of forces, beyond which the active core network becomes more stable. Deformation in the network becomes irreversible beyond a threshold force, which depends on the type of distribution; for a uniform distribution of active dipoles, the irreversibility threshold almost coincides with the critical collapse point, it decreases for the active core system, and is decreased further for the active periphery system. It is shown that irreversibility arises due to plastic deformations in the form of crease formation which is not reversible even after the force is turned off or reversed. The folding pathways depend weakly on the temporal stochasticity of the active links, but are highly sensitive to any defects (missing bonds) in the network. Our findings, therefore, suggest active force localization (or delocalization) as a prime method to dynamically alter the mechanical stability and reversibility of the underlying elastic network.

Figures (9)

• Figure 1: Triangular lattices with different active dipole link distributions: (a) a patch like distribution of active dipoles at the center of the network; (b) active dipoles distributed like a band along the periphery; (c) active dipoles distributed like small patches across the network; (d) uniformly distributed active dipoles. All four types of distribution contain about $\phi=0.2$ fraction of active dipole links highlighted in red.
• Figure 2: Schematic diagram depicting a contractile force dipole acting on two neighboring nodes connected with a spring.
• Figure 3: (a) Time-averaged steady state radius of gyration $R_g$ plotted as a function of force for the active core (ac), active periphery (ap) and active patches (as) networks with $\phi = 0.2$ fraction of dipole links. For comparison we have also shown the data for the uniform distribution (ud), previously reported in Ref. majumdar2025. (b) For forces $f>1$, $R_g$ scales as $(1-\phi)^{\nu}$ with $\nu=2.5$ for different patch sizes containing fraction of dipole links $\phi=0.1-0.5$. (c) Comparison between the relaxation time to reach the steady state for ud and ac at $f=1$. (d) Relative decrease of the active and passive $R_g$ components of the ac network. The radius of gyration $R_g$ has been normalised by the $R_g$ at $f=0.1$. The dashed lines correspond to $f=0.65$ and $f=1$.
• Figure 4: (a) $R_g$ as a function of $\phi$ at a constant fixed active force amplitude $f=1$ for active core (ac) and active periphery (ap). For uniform distribution (ud) the force is fixed at $f=0.65$. (b) Near steady state snapshot for the active core system at a force amplitude $f=3$. The passive and active parts of the network are colored in blue and red, respectively. (c-d) Heatmap plot of the steady-state mean stress $\sigma_{mean}$ including both active and passive components of forces and calculated using Eq. \ref{['eq_sigma_mean']} for the ac network in (c) and ud network in (d).
• Figure 5: (a) Relative decrease of the active and passive components of the active periphery (ap) network. The radius of gyration $R_g$ has been normalised by the $R_g$ at $f=0.1$. The dashed lines correspond to $f=0.45$ and $f=1.2$. (b) Active periphery network size scaling in the high force regime $(f>1)$ with fraction of dipole links $\phi$. $R_g$ scales as $(1-\phi)^{\nu}$ with $\nu=1.75$. (c) Steady state snapshot for the active core system at a force amplitude $f=3$. The passive and active parts of the network are colored in blue and red, respectively. (d) Heatmap plot of the steady-state mean stress $\sigma_{mean}$ at force $f=3$, including both active and passive components of the forces, calculated using Eq. \ref{['eq_sigma_mean']}.