Force generation by a propagating wave of supramolecular nanofibers

Out-of-equilibrium chemical and biological processes frequently produce spatiotemporal patterns such as oscillations and traveling waves, which in living cells drive essential functions including division and migration. Actin waves in lamellipodia generate mechanical forces to advance the cell membrane. Translating such capabilities to artificial systems could endow soft materials with life-like properties. While various dissipative and fuel-driven supramolecular systems have been explored, controlled spatiotemporal coupling of formation and degradation to create traveling patterns remains challenging, and direct, quantitative force measurements in artificial settings are scarce. This study aims to design an artificial reaction network that mimics actin’s spatially regulated polymerization/depolymerization using two orthogonal chemical stimuli to program the formation and degradation of peptide-based supramolecular nanofibers. The authors seek to visualize a propagating wave of nanofibers, identify the key physicochemical parameters governing wave emergence, and quantify the mechanical force generated by such a wave.

Prior work has shown that coupling soft materials to out-of-equilibrium reactions (e.g., Belousov–Zhabotinsky) yields self-oscillating gels, walking actuators, and autonomous transport. Fuel-driven dissipative supramolecular systems demonstrated transient fiber assembly and degradation, but often without spatiotemporal coupling. Traveling fronts of supramolecular colloids have been reported, though degradation was not simultaneous. Collectively, these studies highlight possibilities for pattern formation but lack rational guidelines for waves based on coupled formation and degradation of supramolecular nanofibers and quantitative force readouts. The biological literature on actin/microtubule force generation provides motivation and comparative benchmarks for artificial systems.

Design principles: four guidelines were set to create a propagating wave: (1) drive with spatial gradients of stimuli; (2) use two distinct chemical stimuli to induce formation and degradation; (3) employ a monomer with two orthogonal functional groups; (4) avoid mutual interference between stimuli. System: a peptide-type hydrogelator BPmoc-F3 bearing a carboxylate (Zn2+ coordination site) and a boronobenzyl oxycarbonyl group (H2O2-labile) was used. Formation stimulus: Zn2+ (e.g., Zn(NO3)2) promotes coordination-driven assembly of nanofibers. Degradation stimulus: H2O2 generated by glucose oxidase (GOx)/glucose triggers 1,6-elimination to cleave the boronobenzyl group, decomposing fibers. Orthogonality was ensured since Zn2+ is redox-inert toward the GOx/glucose system. Experimental assays: (i) Bulk sol–gel–sol transitions were probed by tube inversion. Typical conditions: BPmoc-F3 2.4 mM, GOx 1 mg/mL, Zn(NO3)2 1.2 mM (0.5 eq), glucose 4.8 mM (2 eq), 50 mM HEPES pH 7.4, 30 °C. Control additions of water or EDTA verified Zn2+-specific gelation and chelation-induced liquefaction. HPLC quantified monomer decomposition after glucose addition. (ii) Real-time CLSM imaging used a fluorescent probe (BP-TMR) to visualize Zn2+-induced fiber formation and GOx/glucose-triggered degradation. Formation proceeded via stochastic seed nucleation followed by fiber growth into a network over >60 min; degradation was spatially homogeneous over ~60 min when glucose alone was added to pre-formed gels. (iii) Propagating wave assay: a droplet containing BPmoc-F3, BP-TMR, and GOx was sandwiched between glass plates. After 1 min of CLSM observation, a mixture of Zn(NO3)2 (0.5 eq) and glucose (2.0 eq) was injected at the right edge, generating spatial gradients. Time-lapse CLSM tracked fluorescence intensity along x to quantify wave position, duration, distance, and velocity under various GOx/glucose conditions. Simultaneous addition was compared to sequential addition controls. (iv) Reaction–diffusion numerical simulation: a model with variables for nanofiber (n), monomer (m), formation stimulus (x), and degradation stimulus (y) was constructed with linearized formation steps and second-order degradation, and diffusion-only decay of stimuli. PDEs: ∂n/∂t = Dn∇2n + k1mx + k2nm − k3ny + k5nmx; ∂m/∂t = Dm∇2m − k1mx − k2nm − k4my; ∂x/∂t = Dx∇2x; ∂y/∂t = Dy∇2y. Parameter regimes assumed k2 and Dn much smaller than others. Simulations produced spatiotemporal maps and kymographs and analyzed dependencies (e.g., velocity vs kinetics and Dy; role of gradient of y). (v) Force measurement: PEG-coated fluorescent polystyrene beads (500 nm diameter; 20 µg/mL) were dispersed in the droplet. During wave propagation, bead trajectories were recorded by CLSM to extract velocities. Assuming medium viscosity ~2.5 mPa·s, minimal forces were calculated via Stokes drag (F = 3πηνd·v). Controls with homogeneous formation or degradation verified specificity. (vi) Persistence length measurement: separate high-speed CLSM imaging (33 fps, 100×) of Zn2+-induced fibers; image skeletonization and spline interpolation in MATLAB yielded 2D persistence length from angular correlation ⟨cos θ(s)⟩ = exp(−s/L).

• Orthogonal chemical control: Zn2+ triggered rapid nanofiber formation and gelation; GOx/glucose-generated H2O2 decomposed BPmoc-F3 and collapsed fibers, with negligible mutual interference.
• Propagating wave visualization: Simultaneous injection of Zn(NO3)2 (0.5 eq) and glucose (2.0 eq) at a droplet edge produced a traveling band of fiber formation followed by degradation along the x-direction. After passage, fluorescence vanished, indicating complete decomposition.
• Quantitative wave metrics: In wide-field CLSM, the wave propagated over 340 ± 40 µm at an average speed of 54 ± 8 µm/min. The wave onset and peak fluorescence at given x positions exhibited clear temporal delays consistent with propagation.
• Condition dependence: Halving GOx to 0.5 mg/mL still produced waves but with reduced speed (20 ± 2 µm/min) and longer duration. Using only 1.0 eq glucose yielded no wave—fibers formed and then degraded homogeneously, with intensity time courses independent of position. Excess GOx (2 mg/mL) or glucose (63 eq) prevented appreciable fiber formation due to rapid degradation. Sequential addition (glucose 30 min after Zn2+) led to homogeneous degradation without wave, underscoring the need for simultaneous stimuli and appropriate gradients.
• Numerical simulation: Reaction–diffusion modeling reproduced traveling bands of fibers with limited width. Key determinants included (i) concentration gradients of both formation and especially degradation stimulus, (ii) much smaller diffusion coefficient of the nanofibers than other species, and (iii) faster formation kinetics increased wave speed, while higher Dy also increased speed. Initial anisotropic stimulus shapes decayed to half-round distributions and were not essential to the observed crescent wavefront.
• Force generation: Beads moved along the wave direction with average velocity ~0.4 µm/s and displacements ~10 µm during propagation. Using η ≈ 2.5 mPa·s, the minimal force was estimated at ~0.005 pN. Homogeneous formation or degradation did not induce notable bead movement.
• Additional observations: Fiber formation continued for over 60 min under Zn2+; degradation under glucose proceeded homogeneously over ~60 min when not coupled to simultaneous formation. The wavefront shape varied due to manual injection but the phenomenon was reproducible.

The study demonstrates that a rationally designed, orthogonally controlled reaction network can couple supramolecular nanofiber formation and degradation to produce a self-propagating wave at the millimeter scale. The findings address the central question of how to program spatiotemporal patterns in artificial supramolecular systems and whether such patterns can generate mechanical forces. Experiments and simulations converge on the importance of spatial gradients of the chemical stimuli and the comparatively slow diffusion of supramolecular fibers as critical ingredients for wave emergence and maintenance. The measured bead displacements and calculated forces indicate that chemophoresis and depletion-based interactions associated with the traveling fibrous band are sufficient to transport mesoscale objects, drawing a parallel to actin-driven forces in cells. The reaction–diffusion analysis clarifies how tuning formation kinetics and the diffusion of the degradation stimulus modulates wave velocity, suggesting routes to actively control pattern dynamics (e.g., via microfluidics or stimulus dosing). These insights bridge biological pattern-driven mechanics and artificial supramolecular materials, advancing the design of active, life-like soft matter.

This work establishes a design principle for generating traveling waves in artificial supramolecular systems by coordinating orthogonal chemical stimuli that independently control fiber assembly and disassembly. Real-time imaging and modeling reveal that stimulus gradients and markedly lower diffusion of the fiber phase underpin wave propagation. Importantly, the propagating wave generates measurable forces (~0.005 pN) capable of moving nanoscale beads, representing, to the authors’ knowledge, the first quantitative force determination arising from an artificial supramolecular spatiotemporal pattern. These results open avenues to engineer active soft materials for guided transport, actuation, or mechanochemical feedback. Future research could integrate controlled flow fields to tune gradients, explore alternative orthogonal chemistries and stimuli, quantify local rheology within the wave, and couple waves to functional cargos or responsive interfaces to harness and amplify mechanical output.

Wave shape varied between experiments due to manual injection, indicating sensitivity to initial stimulus distribution. The local viscosity within the propagating wave could not be directly measured; force estimates used an upper-bound bulk viscosity (2.5 mPa·s), introducing uncertainty. Wave generation required specific concentrations and simultaneous stimulus addition; outside narrow windows, the system either degraded homogeneously or failed to form fibers, which may limit robustness and generalizability. The reaction–diffusion model linearized formation steps and neglected potential interference between stimuli or side reactions, simplifying real kinetics. Diffusion coefficients, especially of the nanofibers, were assumed rather than directly measured.