Ultrafast reversible self-assembly of living tangled matter Vishal P. Patil1Harry Tuazon2Emily Kaufman2Tuhin Chakrabortty2David Qin3J orn Dunkel4and M. Saad Bhamla2

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Ultrafast reversible self-assembly of living tangled matter

Vishal P. Patil,1, ∗Harry Tuazon,2, ∗Emily Kaufman,2Tuhin

Chakrabortty,2David Qin,3J¨orn Dunkel,4, †and M. Saad Bhamla2, †

1School of Humanities and Sciences, Stanford University, 450 Serra Mall, Stanford, CA 94305

2School of Chemical and Biomolecular Engineering,

Georgia Institute of Technology, Atlanta, GA 30318

3Wallace H. Coulter Department of Biomedical Engineering,

Georgia Institute of Technology, Atlanta, GA 30332

4Department of Mathematics, Massachusetts Institute of Technology,

77 Massachusetts Avenue, Cambridge, MA 02139

(Dated: October 10, 2022)

Tangled active ﬁlaments are ubiquitous in nature, from chromosomal DNA and cilia carpets to

root networks and worm blobs. How activity and elasticity facilitate collective topological trans-

formations in living tangled matter is not well understood. Here, we report an experimental and

theoretical study of California blackworms (Lumbriculus variegatus), which slowly form tangles over

minutes but can untangle in milliseconds. Combining ultrasound imaging, theoretical analysis and

simulations, we develop and validate a mechanistic model that explains how the kinematics of indi-

vidual active ﬁlaments determines their emergent collective topological dynamics. The model reveals

that resonantly alternating helical waves enable both tangle formation and ultrafast untangling. By

identifying generic dynamical principles of topological self-transformations, our results can provide

guidance for designing new classes of topologically tunable active materials.

Filaments and ﬁbers are a crucial building block of

complex matter, giving rise to a broad variety of mor-

phologies with distinct mechanical and topological prop-

erties. From entangled polymeric systems [1–8], active

cilia carpets [9] and worms blobs [10], to everyday macro-

scopic materials including yarn [11, 12], hair [13] and

fabrics [14], the propensity of the underlying ﬁlaments to

tangle [15–17] is responsible for the emergent dynamics

of a range of biological and physical systems. The result-

ing topological obstructions [18–20] induce constraints on

motion that can lead to materials with diﬀerent trans-

port [21, 22], stress-response [23–26] and energetic prop-

erties [22]. Although tangling can inhibit functionality in

common materials [27], the topological control and ma-

nipulation of amorphous tangles has remained a tantaliz-

ing theoretical and experimental challenge. In particular,

the question of how to quickly unravel a complex tangle

presents a historically famous problem [28, 29], of equal

importance to comb makers [30] and coiﬀeurs [13, 31] as

to cells [32] and crawling animals [33, 34].

Living tangled matter, consisting of ﬁlamentary ob-

jects which can braid and wind around each other, rep-

resents an important class of topological active mat-

ter [34–36]. Such systems are often capable of dynam-

ically controlling their topological state, and exploiting

apparently disordered tangles as a resource [10, 34, 37–

39]. A particularly striking example is the California

blackworm (Lumbriculus variegatus) [34], owing to its

ability to assemble into three-dimensional (3D) tangles

over the course of minutes, and rapidly disentangle in

milliseconds (movie S1). Biologically, a blackworm col-

lective uses the tangled state to eﬃciently execute a range

of essential functions, such as temperature maintenance

and moisture retention [10, 39, 40]. Perhaps even more

importantly, the ability to rapidly escape from the tan-

gle is an important predation response [33]. The bio-

physical mechanisms by which basic ﬁlamentous organ-

isms can achieve such ultrafast untangling have remained

unknown. Here, motivated by this question, we com-

bine ultrasound imaging experiments and elasticity the-

ory to explain how individual worm gaits lead to large-

scale topological dynamics and transitions. By map-

ping worm tangling to percolation problems and picture-

hanging puzzles [41], our analysis shows how resonantly

tuned helical waves enable self-assembly and rapid un-

knotting of tangled matter, thus revealing a generic dy-

namical principle that can guide the design of novel active

materials.

Blackworms are capable of assembling into topologi-

cally intricate tangles consisting of anywhere from 5 to

50000 worms [10] (Fig. 1A). Our ultrasound experiments,

conducted on worm tangles immobilized in gelatin, al-

low for the reconstruction of the 3D structure of a living

tangle (Fig. 1B,C; Methods). This reveals a picture of

the tangle as a strongly interacting system, in which the

worms are tightly packed (Fig. 1D), and most worms are

in contact with most other worms (Fig. 1E). In addi-

tion to the arrangement of contact, the non-topological

structure of the worm tangle can also be described by

the variation of geometric quantities both within and be-

tween diﬀerent worms. To analyze the tangle geometry,

we approximate each worm as a curve x(s), parameter-

ized by arc length s, which can be characterized by lo-

cal in-plane curvature, κ(s), and an out-of-plane 3D tor-

sion, τ(s) (SI). These give rise to bending strain, =κh

(Fig. 1F), and chirality, χ=κ2τ(Fig. 1G), where his

arXiv:2210.03384v1 [cond-mat.soft] 7 Oct 2022

A B C

D E

0 10

Distance (h)

-1.0 +1.0

0 1.0

Ultrasound 1

Ultrasound 2

Ultrasound 3

0 5 10

Distance (h)

3D chirality correlation

0 5 10

Distance (h)

3D strain correlation

F HG I

Chirality, (h

-3

)Strain,

FIG. 1. 3D ultrasound data reveal the mechanical structure of active, biological worm tangles (A) Topologically

complex tangle formed by Lumbriculus variegatus consisting of approximately 200 worms. Scale bar 3mm. (B,C) Ultrasound

imaging reveals the interior structure of a 12-worm tangle. Scale bar 5mm. (D,E) The contact matrix and contact graph conﬁrm

that the worm tangle is a strongly interacting system. (F,G) 3D experimental data enable the visualization of strain , and

chirality χ, ﬁelds within the tangle, revealing that the worms form achiral tangles. (H,I) Decorrelation of strain, ρC((x), (y)),

and chirality, ρC(χ(x), χ(y)), over distances of |x−y| ≈ 2.5h(dotted lines) demonstrates the limits of a continuum elastic

theory for worm tangles. The decorrelation length scale indicates the existence of an eﬀective radius, heﬀ ∼1.25h, arising from

the preparation of tangles for ultrasound (Methods).

the worm radius. The 3D distribution of both strain and

chirality is primarily heterogeneous (Fig. 1F,G). More

formally, the correlation coeﬃcients for strain and chi-

rality, ρC((x), (y)) and ρC(χ(x), χ(y)), decay rapidly

as functions of the spatial separation, |x−y|(Fig. 1H,I).

For small values of |x−y|, the correlation functions

are dominated by intraworm interactions, but decorrela-

tion occurs once ρCbegins to include interworm eﬀects.

In particular, ρ≈0 for both strain and chirality once

|x−y|>2.5h, which indicates the existence of an ef-

fective radius, heﬀ = 1.25h. This eﬀective radius is a

signature of the ultrasound protocol (Methods), which

requires the tangles to undergo a small dilation. The

rapid decorrelation demonstrates that strain and chiral-

ity are not described by 3D continuum ﬁelds, illustrat-

ing the diﬃculty of constructing a continuum theory for

the living tangle. Understanding the mesoscale structure

of the tangle requires moving beyond purely geometrical

properties.

Topological analysis of the tangle geometry allows us

to distinguish between diﬀerent forms of contact. The

intuitive notion that worms which braid should interact

more strongly than worms which simply touch can be

captured by considering the linking number [44], Lk, of

Contact link

A B C

0 0.8 1.6

Contact link between two worms

Probability density

0 1 2 3

Tube radius (h)

Total contact link per worm

Ultrasound 1

Ultrasound 2

Ultrasound 3

Ultrasound 1 Ultrasound 2

Ultrasound 3

FIG. 2. Topological structure of worm tangles (A) 3D ultrasound reconstructions (as in Fig. 1B,C) allow for the individual

topological interactions between chosen worms (solid color) to be mapped in detail. Scale bar 5mm. (B) Topological analysis

enables the classiﬁcation of tangle structure by distinguishing between contact (left column) and braiding interactions (right

column), which are deﬁned by having linking number |Lk|>1/2. (C) Contact link, cLk, deﬁned as the absolute value of the

link between worms separated by at most 2heﬀ , identiﬁes the strongest topological interactions within the tangle. The contact

link between non-touching worms is 0. Pairs of worms with cLk > 1/2 are highlighted in red. (D) The tangle graph provides a

sparser representation of tangle state than the contact graph. Edges are present between pairs of worms with cLk > 1/2, i.e.

worms that both touch and have |Lk|>1/2 (red bordered squares in C). (E) The probability distribution of the contact link

between two worms is stable across ultrasound data sets. Pairs of worms with contact link greater than 1/2 (dotted line) lead

to edges in the corresponding tangle graphs (inset), with edge thickness given by the value of the contact link. (F) Increasing

the tube radius of the worm curves modiﬁes the contact structure of the tangle and thus increases the total contact link (SI).

The radius dependence of total contact link is similar across diﬀerent tangles, and indicates the presence an eﬀective radius as

in Fig. 1H,I, that is distinct from the true radius, h.

the i’th worm and the j’th worm

Lkij =1

4πZdsdσ Γij ·(∂sΓij ×∂σΓij ) (1)

where Γij (s, σ) = (xi(s)−xj(σ))/|xi(s)−xj(σ)|, and

xi,xjare the curves representing the i’th and j’th

worms. Although traditionally deﬁned only for closed

curves, the linking number of open curves quantiﬁes en-

tanglement by taking an average of the amount of braid-

ing in every 2D projection [45] (SI). Visually, pairs of

worms with |Lk|>1/2 appear to wind around each other

(Fig. 2A,B). However, Lk is not sensitive to contact,

which must ultimately mediate every worm-worm inter-

action. Accordingly, we deﬁne a more sensitive measure,

termed contact link cLk, by setting cLk =|Lk|for worms

in contact, and cLk = 0 otherwise. In contrast to the con-

tact matrix (Fig. 1D), the contact link matrix (Fig. 2C)

identiﬁes a far smaller number of key interactions, thus

providing a sparser representation of tangle state. This

is particularly evident from the tangle graph (Fig. 2D),

which shows worm-worm interactions with cLk > 1/2.

The robustness of contact link as a measure of tangling is

evident through its behavior across diﬀerent ultrasound

data sets. For example, the probability distribution of

Time (1/α)Time (1/α)

Time (1/α)

0 150

Total contact link per worm

E F

Start

End

-3

-6

610/α50/α90/α130/α

1.6s 8.2s 14.7s 21.3s

Turning angle θ

300 600

Time (ms)

10/α50/α90/α130/α

Turning angle θ

44ms 221ms 398ms 575ms

TanglingUntangling

Time (s)

0 150

Time (1/α)

Time (s) 24.5

0 150

Time (ms) 664

0 50 100 150

Simulations

FIG. 3. Resonant helical worm head dynamics give rise to numerically reproducible weaving and unweaving

gaits. (A,B) Experimentally observed worm head trajectories [42, 43] projected into 2D can be approximated by their angular

direction, θ(t) = arg ˙

x(t), in both the tangling (A) and untangling (B) cases (movie S2). θis characterized by an average

turning rate, α=h| ˙

θ|i, and a rate of switching from left turning (red points, ˙

θ > 0) to right turning (blue points, ˙

θ < 0). The

chirality number, γ=α/2πλ, captures the diﬀerence between weaving (γ= 0.68) and unweaving (γ= 0.36) gaits. α−1deﬁnes

an intrinsic timescale for tangle assembly and disassembly. Scale bars 3mm. (C,D) Experimentally measured head trajectories

of 3 worms (diﬀerent colors) executing the tangling (C) and untangling (D) gaits demonstrate the formation (C) or removal (D)

of topological obstructions within a similar time in units of α−1. Scale bars 5mm. (E) Simulations of active Kirchhoﬀ ﬁlaments

demonstrate that the gaits described in (A,B) are suﬃcient for reversible tangle self-assembly (movie S2). The topological

state is quantiﬁed using tangle graphs (inset). Tangling ﬁlaments have large γ(top row E and A) and untangling ﬁlaments

have small γ(bottom row E and B). The initial tangled state (bottom row E) is obtained from 3D ultrasound reconstruction.

Average worm lengths range from 40 mm (top row) to 28 mm (bottom row), with radius 0.5 mm throughout. Displayed worms

are thickened to aid visualization. (F) The total contact link per worm (Fig. 2) obtained from simulations reveals the rate at

which tangles form (purple dots, top row panel E) and unravel (green dots, bottom row panel E).

the contact link between two worms, a measure of topo- logical interaction strength, retains a characteristic shape

for all three tangles (Fig. 2E). Additionally, the total con-

tact link (SI), obtained by summing all the pair contact

links from Fig. 2C, is sensitive to the contact structure

of the tangle. In particular, the total contact link as a

function of worm radius (Fig. 2F) behaves similarly as

the worms are thickened from zero radius to larger radii.

Thus, by incorporating topological information [45, 46]

as well as geometric information, contact link cLk cap-

tures core structural motifs that are reproducible over

diﬀerent experiments. In particular cLk will enable us

to compare experimentally observed worm tangles with

tangled structures generated by dynamical simulations.

The ability of the blackworm to form tangles over

minutes (Fig. 3A), but rapidly unravel in milliseconds

(Fig. 3B) is a key biological and topological puzzle [37,

38]. To understand the dynamical process that gives rise

to tangle formation, we experimentally studied the head

trajectories of single worms (Fig. 3A-D; Methods). Since

these experiments were performed in a shallow ﬂuid well

(height ∼2 mm), the projection of the trajectories into

2D (Fig. 3A-D) does not cause signiﬁcant information

loss. To capture the winding motions associated with

braiding and unbraiding, we assume the worm head has

preferred speed v=h| ˙

x(t)|i, and focus on the worm turn-

ing direction, θ(t) = arg ˙

x(t). The θtrajectories can be

approximately described in terms of two parameters, the

average angular speed α=h| ˙

θ|i (Fig. 3A,B) and the rate

λat which ˙

θchanges sign. These quantities can be esti-

mated from the noisy trajectory data (SI). Although the

characteristic timescales α−1for slow tangling and ultra-

fast untangling diﬀer by 2 orders of magnitude, rescaling

the θtrajectories for each gait by α−1reveals a similar

underlying dynamics (Fig. 3A,B). This similarity reﬂects

the fact that locomotion machinery is biologically con-

strained [47], and indicates that tangling and untangling

can be captured by the same mathematical model. To

conﬁrm this, we ﬁrst formulate a minimal 2D model of

worm head dynamics which we will then generalize to a

full 3D dynamical picture.

A minimal 2D model can be constructed by focusing

on the helical worm head dynamics identiﬁed experimen-

tally (Fig. 3). In particular, the quantities α, λ and v

discussed above motivate the following stochastic diﬀer-

ential equation (SDE) model for a worm-head trajectory

x=vnθ+ξT,˙

θ=σ(t;λ)α+ξR(2)

where ξT, ξRare noise terms, nθis a unit vector in the

θdirection and σ(t;λ) switches between +1 and −1 at

rate λ(SI). These trajectories can be further classiﬁed

by dimensionless parameters. In particular, the chirality

number, γ=α/2πλ, distinguishes between the tangling

and untangling gaits (Fig. 3A,B). This non-dimensional

parameter corresponds to the average number of right-

handed or left-handed loops traced out by the worm

before changing direction and provides an intuitive way

of understanding the topological properties of each gait.

When γis large, worms wind around each other before

switching direction, thus producing a coherent tangle.

On the other hand, for small γ, the worms change di-

rection before they are able to wind around one another

and so remain untangled. Our trajectory model thus ex-

plains how the characteristic helical waves produced by

untangling worms mediate topology (movie S2).

We next show that these conclusions generalize to a

full 3D mechanical model of worm gaits. To model the

worms, we performed elastic ﬁber simulations where the

worms are treated as Kirchhoﬀ ﬁlaments [48–55] with

active head dynamics (SI). The head motions are pre-

scribed by the SDE model (2) together with additional

3D drift (SI). The resulting worm collectives can form 3D

tangled structures (Fig. 3E) consistent with those seen in

our experiments, as quantiﬁed by contact link (Fig. 3F).

In particular, the tangling and untangling behavior in

these simulations appears to be a function of the chirality

number, γ, further conﬁrming its importance (Fig. 3E,F;

movie S2). This formulation of a 3D dynamical model al-

lows us to understand how the dynamics of single worms

produces worm collectives with distinct topologies.

Based on the above analysis of the worm trajecto-

ries we can build a mean-ﬁeld tangling model, which es-

tablishes a mapping between tangling and percolation

(Fig. 4). To formulate an analytically tractable model,

we treat the worm motion as approximately 2D, so each

worm eﬀectively moves in a 2D slice of the 3D tangle

(Fig. 4A,B). As a given worm moves in a plane P, its

head traces out a curve, x(t) (Fig. 4B, purple and green

curves), described by equation (2). The worm can en-

counter obstacles, which represent intersections of the

other worms with the plane P(Fig. 4B, colored cir-

cles). For simplicity, we treat these obstacles as forming

a square lattice Λ ⊂P, with spacing `(Fig. 4B), which

corresponds to the tightness of the worm tangle (SI). The

3D notion of contact link between worms can be mapped

to this 2D picture [41] by considering the winding of the

trajectory x(t) around the obstacles p∈Λ. In particular,

let Wpbe the winding number of x(t) around p∈Λ after

time t=L/v, the time taken for the worm-head to move

one worm length, L(SI). The contact winding, cWpof

x(t) around pis then |Wp|if x(t) gets within a thresh-

old distance of p(SI), and is 0 otherwise (Fig. 4B). As

observed in experiments (Fig. 3), topological properties

such as the contact winding are sensitive to the chirality

number γ(Fig. 4B). Thresholding and averaging all the

contact winding numbers yields a tangling index

T=*X

p∈Λ

Θ (cWp−1)+(3)

where the step function Θ returns 1 if cWp>1 and 0 oth-

erwise. The tangling index therefore counts the number

of obstacles that a worm winds around (Fig. 4B), and is a

measure of the mean degree of a tangle graph. Since con-

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Ultrafastreversibleself-assemblyoflivingtangledmatterVishalP.Patil,1,HarryTuazon,2,EmilyKaufman,2TuhinChakrabortty,2DavidQin,3JornDunkel,4,yandM.SaadBhamla2,y1SchoolofHumanitiesandSciences,StanfordUniversity,450SerraMall,Stanford,CA943052SchoolofChemicalandBiomolecularEngineering,GeorgiaInstitute...

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Ultrafast reversible self-assembly of living tangled matter Vishal P. Patil1Harry Tuazon2Emily Kaufman2Tuhin Chakrabortty2David Qin3J orn Dunkel4and M. Saad Bhamla2

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