Untangling Topology with California Blackworms

Elastic filaments have a tendency to spontaneously tangle, as evidenced by the frequent development of knots in electrical cords, headphones, and garden hoses. However, another class of systems can detangle just as rapidly without conscious intervention. Biological polymers inside cells—such as DNA, RNA, proteins, and other complex molecules—primarily reside in a snarled-up state but can spontaneously organize themselves to perform essential processes, particularly cell division.

California blackworms (Lumbriculus variegatus) exhibit a macroscopic version of this phenomenon, forming spheroidal tangles that consist of anywhere from five to 50,000 individual worms (see Figure 1). Though completion of this tangling operation can take from several minutes to as long as a half hour, detangling is practically instantaneous. Because blackworm brains only contain a few neurons, the detangling process must be purely biophysical in nature — analogous to the biopolymers inside a cell.

The relatively large size of these worms—which range from about four to 10 centimeters (roughly 1.5 to four inches) in length—makes them a prime study candidate for researchers who are interested in the mathematics of real-world knots, which have physical properties that affect their topological and geometrical structures. On a practical level, these types of systems can help resolve protein folding problems or lead to the creation of new materials that avoid excessive tangling.

“Lots of materials are made up of tangled filaments,” Vishal Patil, a mathematician in the Department of Bioengineering at Stanford University, said. “Things like polymer gels and—at larger scales—wool, fabrics, and cellulose are all comprised of tangled filaments.” 

Patil’s doctoral research explored the mathematics of physical knots, such as those in shoelaces. This work caught the eye of Saad Bhamla, a biomaterials scientist at the Georgia Institute of Technology who was studying blackworms as a potentially intriguing system for the development of analogues for other tangled-filament materials. Patil and Bhamla’s subsequent collaboration resulted in a paper that recently published in Science [1], wherein Bhamla’s group provided experimental measurements of blackworm tangles to reconstruct their full three-dimensional structure. This structure then served as the basis for the mathematical model. “The fact that Saad and Harry [Tuazon], his grad student, discovered this amazing tangling-untangling behavior in these worms suggests something about tangles that can apply to other contexts as well,” Patil said.

Oh What a Tangled Worm We Weave

Blackworms form tangles to control their temperature, prevent themselves from drying out, and move efficiently as a group. Their intricately constructed clumps can redisperse under threat in a matter of milliseconds. Such behavior indicates that the way in which they form a tangle is essential to the rapidity of the untangling process (in contrast to extension cords’ seemingly spontaneous ability to hopelessly tie the lawnmower to the bicycles¹).

To model the system, the researchers first immobilized a worm clump by embedding it in gelatin and scanning it with an ultrasound. Drawing from knot theory, they defined the linking number between worms as

\[Lk_{ij}=\frac{1}{4\pi}\int\Gamma_{ij}\cdot(\partial_s\Gamma_{ij} \times \partial_\sigma\Gamma_{ij})dsd\sigma,\]

where the matrix

\[\Gamma_{ij}=\frac{\boldsymbol{x}_i(s)-\boldsymbol{x}_j(\sigma)}{|\boldsymbol{x}_i(s)-\boldsymbol{x}_j(\sigma)|}\] involves the vector difference between curves \(\boldsymbol{x}_i\) and \(\boldsymbol{x}_j\) that respectively describe worms \(i\) and \(j\). To reduce the system’s complexity, the modelers defined the contact link as \(cLk_{ij}=|Lk_{ij}|\) if the worms are in contact and \(cLk_{ij}=0\) if they are not touching. They found that most worms had physical contact with nearly every other individual in the gelatin, thus indicating that strong interactions govern the system (see Figure 2).

“We want to understand how you can control the tangled topological state just by looking at the motion of individual filaments,” Patil said. “We built a minimal mathematical model to explain how filament motion can give rise to these different topological states, which means that [the model] is hopefully generalizable to a lot of other filament systems.”

Strictly speaking, mathematical knot theory only describes the topology and geometry of string-like shapes that form closed loops. However, scientists who work with open filaments like shoelaces, worms, or DNA can still utilize many concepts from knot theory, including essential topological features and geometrical measures. In the blackworm scenario, dramatic changes in the linking number describe a topological phase transition from the untangled to tangled state and back again.

As the Worm Turns

To investigate the system’s dynamics, the experimental team placed the worms in a shallow fluid and tracked their trajectories as they formed and unformed clumps. Giving one worm a very gentle electric shock caused the entire clump to unravel, revealing the way in which the “message” to untangle spread amongst the group. “[The worm] starts to do a figure eight motion where it forms a clockwise loop, then an anticlockwise loop, and then a clockwise loop,” Patil said. This wavelike motion passes down the worm’s body and alerts the other worms in the tangle that they should start wiggling free as well.

Patil realized that he could mathematically simplify the motion by reducing it to two dimensions and treating out-of-plane motion projectively. He also tracked the trajectory of just the head of each worm, which allowed him to simulate the system via a set of simple coupled stochastic differential equations. Focusing on the direction in which the worm turns—represented by \(\theta(t)=\arg\dot{\boldsymbol{x}}(t)\), where \(\boldsymbol{x}\) is the two-dimensional position of the worm head—the stochastic differential equations are as follows:

\[\dot{\boldsymbol{x}}=\upsilon\boldsymbol{n}_\theta+\boldsymbol{\xi}_T\] \[\dot{\theta}=\sigma(t;\lambda)\alpha+\xi_R,\]

with noise terms \(\{\boldsymbol{\xi}_T,\xi_R\}\). One can estimate the time-averaged velocities \(\upsilon=\langle|\dot{\boldsymbol{x}}(t)|\rangle\) and \(\alpha=\langle|\dot{\theta}|\rangle\) from experimental data, and \(\sigma(t;\lambda)\) switches between \(+1\) and \(-1\) at rate \(\lambda\). The dimensionless chirality parameter \(\gamma=\alpha/2\pi\lambda\) is inversely proportional to the number of direction changes; small \(\lambda\) indicates that many direction changes occur and the blackworm cluster subsequently detangles.

“When you switch direction rapidly, you untangle,” Patil said. “And when you turn in the same direction for longer, you tangle.” He cautioned that the simplified model is not a full mathematical description of blackworm untangling, though its straightforward explanation might be applicable to other systems.

“What’s quite interesting is that our ability to [observe] the structure of tangled objects remains kind of limited,” Patil said. “If we have a tangled ball of wires, it’s very hard to extract just a single wire. Worms obviously do a lot of stuff when they’re tangling and untangling in a complex biological system, [and] this is a simple model of one motion. There’s a very subtle difference in the transition between these two [states]. But if you can explain it mathematically, then you know why it works.”

In particular, this research helps to elucidate the reason why many non-biological systems can tangle more easily than they untangle, based on the types of motion that are involved in the snarl. A cluster of electrical cords does not spontaneously produce the wavelike motion that blackworms exhibit, but the molecules inside cells—which undergo the kinds of chemical stimuli that govern biological processes—very well might.

¹ A point first made by the late Terry Pratchett in his 1992 novel, Lords and Ladies.

References

[1] Patil, V.P., Tuazon, H., Kaufman, E., Chakrabortty, T., Qin, D., Dunkel, J., & Bhamla, M.S. (2023). Ultrafast reversible self-assembly of living tangled matter. Science, 380(6643), 392-398.