Kinetic Catalysis of Spontaneous Knotting: How Free Particles Modulate Filament Entanglement

Abstract

Entangled knots form spontaneously in flexible filaments, yet the influence of the surrounding environment on this process is poorly understood. Here we demonstrate that free-moving particles act as kinetic catalysts for spontaneous knotting. Through controlled agitation experiments, we find that a small number of inert beads substantially enhance the probability and accelerate the rate of knot formation. This catalytic effect is non-monotonic: an optimal particle size and concentration that maximizes entanglement, while an excess of particles suppresses knotting by impeding the filament's dynamics. We develop a stochastic model that quantitatively reproduces this behavior, attributing it to a competition between entanglement-promoting collisions and motion-suppressing drag. Our findings reveal a mechanism for tuning topological complexity, whereby adjusting these environmental agitators can either promote rapid self-assembly or inhibit unwanted entanglement. This work suggests new strategies for controlling filament topology in settings ranging from crowded biological environments to advanced materials processing.

Figures (4)

• Figure 1: Probability of spontaneous knot formation as a function of string length $L$. Data are shown for strings without any beads (blue circles) and with two 15 mm silicone beads present as free agitators in the container (red circles). Error bars represent the 95% Wilson score confidence interval. Solid lines are fits to the kinetic model $P(L) = P_{\text{max}}(1 - \exp[-(L/L_c)^\alpha])$, quantifying the significant enhancement in knotting probability in the presence of the beads.
• Figure 2: Kinetic model of spontaneous knot formation and the resulting knotting rate. (A-C) Schematic representation of knot formation via the threading of a mobile string end (path indicated by dashed line) through loops formed by the string body. (A) A path involving multiple crossings that is topologically equivalent to the unknot. (B) A path that forms the simplest prime knot, the trefoil ($3_1$), by threading a self-generated loop. (C) A more complex path that threads a subsequent loop, resulting in a higher-order knot ($6_2$). (D) The length-dependent knotting rate, $k(L)$, derived from fitting the experimental data in Fig. \ref{['fig:1']} to the kinetic model.
• Figure 3: Proportion of trefoil ($3_1$) knots among all knotted outcomes as a function of string length. The decrease in this proportion with length indicates an increase in knot complexity. The data for conditions with and without beads are statistically indistinguishable, suggesting the beads do not alter the complexity of the knots formed.
• Figure 4: Non-monotonic influence of bead parameters on knotting probability. (A) Knotting probability relative to the no-bead baseline, as a function of the number of 15 mm beads in the container. (B) Relative knotting probability for two beads as a function of bead diameter. The relative probability is the ratio of the knotting probability with beads to that without beads. An optimal enhancement is observed in both cases.