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Pattern formation and polarity sorting of driven actin
filaments on lipid membranes

Alfredo Sciortinoa,b and Andreas R. Bauscha,b,1

aLehrstuhl für Biophysik (E27), Technische Universität München, D-85748 Garching, Germany; and bCenter for Protein Assemblies, 85747
Garching, Germany

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved January 3, 2021 (received for review August 13, 2020)

Collective motion of active matter is ubiquitously observed, rang-
ing from propelled colloids to flocks of bird, and often features the
formation of complex structures composed of agents moving
coherently. However, it remains extremely challenging to predict
emergent patterns from the binary interaction between agents,
especially as only a limited number of interaction regimes have
been experimentally observed so far. Here, we introduce an actin
gliding assay coupled to a supported lipid bilayer, whose fluidity
forces the interaction between self-propelled filaments to be dom-
inated by steric repulsion. This results in filaments stopping upon
binary collisions and eventually aligning nematically. Such a binary
interaction rule results at high densities in the emergence of dy-
namic collectively moving structures including clusters, vortices,
and streams of filaments. Despite the microscopic interaction hav-
ing a nematic symmetry, the emergent structures are found to be
polar, with filaments collectively moving in the same direction.
This is due to polar biases introduced by the stopping upon colli-
sion, both on the individual filaments scale as well as on the scale
of collective structures. In this context, positive half-charged topo-
logical defects turn out to be a most efficient trapping and polarity
sorting conformation.

active matter | self-propelled rods | ordering phenomena |
high-density gliding assay | actin motility assay

Collective motion of active matter is ubiquitous, with obser-

vations ranging from flocks of birds (1) and schools of fish
(2) to propelled colloids (3). The interactions between agents in
such systems lead to the formation of complex structures in-
cluding clusters, swirls, or lanes of agents moving coherently (4).
The structure of the emerging patterns strongly depends on both
the agents’ shape and their velocity alignment mechanism. A
particular case is that of elongated microscopic particles that
translate along their major axis in a quasi-two-dimensional en-
vironment and only interact upon collision (5, 6). In the context
of cytoskeletal systems, gliding actin filaments or microtubules
propelled by molecular motors are found to be able to readily
crawl over each other and only retain a weak level of alignment
upon binary collisions, which eventually leads at high densities to
a diverse set of patterns (7). Such resulting patterns are found to
be strongly dependent on this weak microscopic alignment in-
teraction, and therefore, even slightly tuning it causes the system
to switch between polar and nematic phases, separated by a
coexistence regime (8, 9). Observed structures in cytoskeletal
systems with weak to moderate interactions include nematic
lanes, polar waves, and vortices (10–12). Conversely, pattern
formation in systems of elongated bacteria or granular matter is
often based on hard interactions with a strong steric component
(13–18). In this repulsion-dominated regime, particles are unable
to crawl over each other and must stop upon collision. In the
limiting case of spherical self-propelled particles, this kind of
steric interaction can lead to a stable phase separation between
stuck and moving particles, the so-called motility induced phase
separation (MIPS) (19). On the other hand,
in the case of
elongated particles, steric effects can still act as velocity aligning
mechanisms. As orientation mismatches are unstable, particles

end up aligning and this leads to flocking behavior rather than to
phase separation (5, 20–25). This latter case, in which strong
steric constraints dominate binary interactions but alignment is
still present, is poorly understood, and how modeling has to be
extended to account for the emergent collective behavior of
elongated, flexible agents with volume exclusion also remains
still under debate (26–30). This is partly due to the lack of mi-
croscopic experimental systems allowing to explore this regime.
Semiflexible cytoskeletal filaments would be the best candidate,
but their volume exclusion is usually too weak. However, having
them propelled by motors immobilized on a fluid membrane
would be a promising route to bridge this experimental gap (31).
Here, we enforce a steric repulsion-dominated interaction,
leading to alignment between actin filaments by coupling myosin
motors to a fluid-supported lipid bilayer. Because of the slippage
of the motors on the membrane, the force propelling the fila-
ments is too weak to enable filaments to crawl over each other
and thus effectively implements a repulsion-dominated regime,
with filaments stopping upon collisions. Eventually, however,
because of the thermal fluctuations of their tips, filaments can
align and resume motion. The experimental realization of such a
microscopic binary interaction, based on volume exclusion, en-
ables us to observe and quantify the resulting pattern formation
process in a system of active semiflexible filaments. We then first
characterize the interaction at the single filament scale, showing
that it leads to nematic alignment. As the filaments’ density is
increased, patterns of collective motion emerge, ranging from
clusters to thick streams and vortices. Despite the nematic col-
lision rule, we find the emerging structures to be locally polar.

Significance

Pattern formation processes in active systems give rise to a
plethora of collective structures. Predicting how the emergent
structures depend on the microscopic interactions between the
moving agents remains a challenge. By introducing a high-
density actin gliding assay on a fluid membrane, we demon-
strate the emergence of polar structures in a regime of nematic
binary interactions dominated by steric repulsion. The transi-
tion from a microscopic nematic symmetry to a macroscopic
polar structure is linked to microscopic polarity sorting mech-
anisms,
including accumulation in wedge-like topological
defects. Our results should be instrumental for a better un-
derstanding of pattern formation and polarity sorting pro-
cesses in active matter.

Author contributions: A.S. and A.R.B. designed research; A.S. performed research and
analyzed data; and A.S. and A.R.B. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

This open access article is distributed under Creative Commons Attribution-NonCommercial-
NoDerivatives License 4.0 (CC BY-NC-ND).

1To whom correspondence may be addressed. Email: abausch@mytum.de.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/
doi:10.1073/pnas.2017047118/-/DCSupplemental.

Published February 3, 2021.

PNAS 2021 Vol. 118 No. 6 e2017047118

https://doi.org/10.1073/pnas.2017047118 | 1 of 8

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The repulsion-dominated interaction indeed introduces a polar
bias not only due to the tendency of active filaments or clusters
to keep moving together after a polar collision but also by forcing
filaments with similar orientation to stop and accumulate when
encountering an obstacle. In particular, at high densities, such an
interaction leads to the formation of transient local +1/2 topo-
logical defects, which act as wedges and, therefore, effectively
trap and polarity-sort motile filaments. We interpret this trap-
ping mechanism as an analog of MIPS for elongated self-
propelled particles.

Results
Binary Interactions Have Nematic Symmetry. The actin gliding assay
(32, 33) has been used to study the collective behavior of elon-
gated filaments at high density (34, 35). Here, we bind bio-
tinylated heavy meromyosin (HMM) motors via streptavidin to a
supported lipid bilayer (SLB) containing 0.5% M/M biotinylated
lipids (36) (Materials and Methods). The molecular motors are
thus free to move on the fluid bilayer. The SLB’s lateral diffusion
coefficient was found to be D = (3.4 ± 0.15) μm2/s (mean + SD,
n = 14), as measured by fluorescence recovery after photo-
bleaching (FRAP) according to refs. 31, 37 (SI Appendix, Fig. S1
and Movie S1). Successively, a mixture containing small
phalloidin-stabilized fluorescent actin filaments (mean length L
∼2 μm, geometrical aspect ratio of ∼1:200) at a given concen-
tration is incubated for 5 min. The incubation step is performed
in ATP-free conditions to allow filaments to bind to the motors
in the rigor state. A total of 0.2% methylcellulose is also present
in the chamber to increase the motors’ processivity (8) but in a
concentration low enough to avoid bundling or substantial
filament–filament interaction (38). Finally, ATP as a fuel
is
added to the observation chamber and the surface of the sample
is observed by total internal reflection fluorescence (TIRF) mi-
croscopy. In these conditions the system is active for at least 2 h.
To monitor the motion of individual actin filaments, we first
incubated only a small amount (10 nM) of
filaments and
obtained a final surface concentration of ρ ∼0.08 filaments/μm2
measured right after ATP addition. Freely moving filaments
have a speed of ∼150 nm/s (Movie S2), which is one order of
magnitude slower than observed in a regular HMM motility as-
says on glass surface (1 to 5 μm/s) (35, 39, 40). This is due to the
fact that motors are bound to a fluid bilayer, and thus, slippage
of motors in the membrane limits their efficiency in transmitting
force to the filaments. For the same reason, the motors’ ability to
push filaments against a load is hindered (Fig. 1A) (31). In the
case of a side collision between two driven filaments, we observe
that the one, which has its direction of motion obstructed by the
second one, does not glide over it. Instead, it stops until either
the other filament moves away and thereby the obstacle is re-
moved or until, thanks to the filaments’ flexibility, both filaments
become aligned by fluctuations in the filaments’ tip and can
move again (Fig. 1B and Movie S3). The net effect in individual
collisions is thus an effective alignment. It occurs over ∼30 to
60 s (Fig. 1D), and its outcome has a dependency on the angle
between filaments at which the collision takes place. We mea-
sured the angle between the filaments’ directions of motion both
right before (θin) and right after (θout) collision occurs (Materials
and Methods). We found that in 90% of the cases the final angle
is either ∼0° or 180°, with collisions at small (large) angles
(i.e., θin < 90° [θin > 90°]) resulting in a final angle of θout ∼0°
(θout ∼180°) (Fig. 1C). Thus, on average, alignment can be either
polar, with filaments moving together after having collided, or
antipolar, with filaments moving in opposite directions. Colli-
sions happening at a perpendicular angle result in a perfectly
nematic response of the outgoing angles. Only a fraction (∼5%)
of the collisions, mostly occurring between perpendicular fila-
ments, does not lead to an alignment as the obstacle glides away
too soon. While right after antipolar collisions the filaments glide

in different directions, the tendency of filaments to move to-
gether for a prolonged time when colliding polarly introduces an
asymmetry at the binary collision level, which stabilizes polar
pairs with respect to antipolar ones (5). Additionally, as fila-
ments stop upon collision, it is possible that multiple filaments
aggregate transiently against the same obstacle. As all such fil-
aments are oriented in the same direction, this further increases
the tendency toward the formation of transient polar order
(Fig. 1E). Thus, despite the microscopic nematic symmetry of the
binary interaction, the steric repulsion between filaments to-
gether with the stopping behavior introduces a tendency toward
polar alignment, especially in multiparticle collisions, which is
absent in the standard actin gliding assay.

transient alignment of

Emergence of Polar Order at Low Filament Densities. We then in-
creased the surface concentration of filaments. At any concen-
tration,
filaments is observed. After
increasing the surface density of filaments to about ρ ∼ 0.18
filaments/μm2, clusters of filaments moving together appeared.
This concentration is about two order of magnitudes lower than
the critical density for ordering of 5 filaments/μm2 determined in
high-density motility assay on glass (35). Thus, in the repulsion-
dominated alignment regime, collisions of a small number of
filaments are already sufficient to account for the emergence of
local order. By multiplying the surface density ρ of filaments
times the squared mean length L of the filaments, one obtains
the dimensionless parameter ρL2, indicating the scaled number
density (41, 42). While for ρL2 ∼0.32 (ρ ∼0.08 filaments/μm2),
no clustering is visible (Movie S2), at ρL2 ∼0.72 (ρ ∼0.18 fila-
ments/μm2), filaments already aggregate (Fig. 2A). Such a value
is much lower than the Onsager threshold of ρL2 ∼4.7 for the
isotropic–nematic transition (43, 44) of passive filaments, indi-
cating that activity lowers the aggregation threshold in this re-
gime. These values are comparable with the clustering threshold
found by simulations of propelled rods (41, 42, 45, 46) and ex-
periments with bacteria (14), despite the actin filaments here
having a much bigger aspect ratio.

We further increased the surface density of filaments to ρ
∼0.40, 0.55, 0.64, and 0.83 filaments/μm2 (ρL2 ∼1.6, 2.2, 2.56, and
3.32), which resulted in increasingly stable collective patterns
(Fig. 2A and Movies S4 and S5). We also note here that such
surface densities roughly correspond to a 0.8 to 1.7% surface
area coverage.

We further observed that successive individual collisions be-
tween filaments lead to the formation, at short times, of small
polar clusters or “seeds”, composed of a few filaments moving in
the same direction (Fig. 2B). Depending on the local concen-
tration, these can either disassemble or act as seeds for the
further alignment of colliding filaments (Fig. 2C). Being com-
posed of filaments, seeds follow the same dynamics (i.e., they
stop and align upon collision). As seeds colliding at small angles
(<90°) often exit the collision polarly oriented, they have the tendency to travel together and form longer-lived structures. During such collisions, seeds are temporarily immobile and can act as obstacles for other filaments, which then accumulate locally. Ultimately, successive merging events lead to the formation of dynamic streams of filaments, which are elongated structures composed of motile filaments (Fig. 2D and Movie S6). Already at ρ ∼0.18 filaments/μm2, most seeds are stable enough to result in the formation of short streams, while at ρ ∼0.40 filaments/μm2 and above (ρL2 > 1.60), streams are longer and persistent in time,
again confirming a transition to stable order in the range
(ρL2 = 1 − 2) predicted by simulations of propelled rods (41, 45).
This is further characterized by plotting the histogram of the
local density over time (Fig. 2G and Materials and Methods). A
transition is visible at concentration higher than ρ ∼0.08 fila-
ments/μm2 from an initially uniform state to a final state in which

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On glass:

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θin

θout

θin

θout

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Fig. 1. Microscopic interaction between filaments. (A) Schematics of the system. Contrary to what happens on glass, motors on a supported membrane are
incapable of pushing filaments against an obstacle as they slip on the membrane. On glass, conversely, colliding filaments can overcome obstacles and crawl
over each other. (B) Example of a collision between filaments on a membrane resulting in alignment in roughly a minute. (Scale bar, 5 μm). The Inset shows a
schematic of the collision and of the incoming angle θin and the outcoming angle θout. (C) Outcoming angle (θout) of collisions between filaments as a function
of the incoming angle (θin). The dashed line indicates the case of no interaction. Red indicates collisions resulting in polar alignment and blue antipolar. (D)
Angles between four different colliding filaments as a function of time. Insets show different phases of the collision: because of volume exclusion, filaments stop,
fluctuate, and eventually align, either polarly or anti-polarly. (E) Example of a multifilament collision leading to polarity sorting. Filaments acting as an obstacle also act
as accumulation sites. As all filaments accumulating are oriented in the same direction, this introduces a polar bias in the system. (Scale bar, 2 μm).

big areas of the sample are devoid of filaments, whereas the tails of
the distribution expand toward higher values. Long tails are already
present at ρ ∼0.18 filaments/μm2, and the effect is more pro-
nounced as the surface density is increased (SI Appendix, Fig. S2).
Streams move and change shape and direction over time. They
can also merge further, giving rise to either a thicker stream or to
two adjacent streams of filaments moving in opposite directions.
Depending on the surface density, streams can also bend or
merge into a vortex (Fig. 2E and Movie S6), or they can disas-
semble again into smaller streams. As the bulk density is further
increased, streams become thicker, longer, and more persistent
in time. By comparing the transverse intensity profile of indi-
vidual filaments with that of streams, we estimate the thickness
of streams in terms of the number of filaments (SI Appendix, Fig.
S2). While at ρ ∼0.07 filaments/μm2, filaments move individually,
with only transient clusters composed of 2 to 3 filaments
appearing, the average number of filaments in a stream increases
as the surface density increases, with streams composed of up to
100 aligned filaments at ρ ∼0.83 filaments/μm2. Streams thus get
thicker and longer (SI Appendix, Figs. S3 and S4) and disas-
semble less frequently. At the same time, isolated seeds become
less favorable. At all concentrations, individual filaments can
also be observed leaving and joining streams or moving freely in
low density areas. Seeds keep forming, but if the concentration is
high enough, they ultimately either disassemble or join a stream.
Erosion and accretion of streams can only happen on the sides,
as filaments flowing inside the stream are trapped by neighboring
filaments. The steric component of the interaction here also
plays a role in impeding that streams cross each other, forcing
them to either stop or merge upon contact. For this reason,
vortices, when formed by a stream that loops on itself, can be
particularly persistent in time, as filaments composing it are
trapped in their own loop. Vortices are observed with radii 1.5 to
8 μm (and thus with a circumference of 10 to 50 μm), which is
much longer than the average filament’s size of 2 μm. This rules
out that vortices are stabilized by longer individual filaments, as

previously observed in simulations of filaments (47) and bacterial
systems (48). It rather indicates that vortices are formed by
streams and seeds colliding and merging, as previously observed
in simulations of semiflexible filaments with hard repulsion (26),
and that they are always composed by multiple filaments. Vortices
are thus, due to the repulsion-dominated interaction that allows
filaments to form structures, longer than their individual size.

To further understand the formation of collective structures,
we computed the histogram of curvature radii, separately for
vortices and streams, at different surface densities (Fig. 2H and
Materials and Methods). The distributions appear similar for all
densities, with the mean curvature radius only slightly increasing
with the density. Vortices appear to be able to bend more than
streams and thus can have a smaller radius of curvature, sug-
gesting that highly bent streams often end up in a vortex. Ad-
ditionally, as no vortex is observed with a mean radius below
1.5 μm, we conclude that vortices can start forming only when
the streams’ length is comparable to the minimum vortex cir-
cumference (∼10 μm) or enough streams and seeds are present
that can loop on each other. No vortices are indeed observed at
ρ ∼0.18 filaments/μm2, when only a few such long streams are
present (SI Appendix, Fig. S3), while they appear at ρ ∼0.40 fil-
aments/μm2 and increase in number at higher densities (Fig. 2A).
At higher densities, most of the filaments are eventually part
of a dense structure, either a stream or a vortex. The filaments,
as well as the formed structures, are all still mobile with a fre-
quent exchange of the filaments with the surrounding less dense
phase. The speed of filaments inside streams and vortices in the
absence of obstacles is around 50 nm/s and thus only weakly
affected by the local density. All observed collective structures
(seeds, streams, and vortices), being composed of filaments in-
capable of crawling over each other, still stop upon collision with
an obstacle. Several events of stopping, aggregating, and then
moving as a coherent polar structure are observed also at the
level of seeds and streams (Fig. 2 C, D, and F). As a result of

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Pattern formation and polarity sorting of driven actin filaments on lipid membranes

PNAS | 3 of 8
https://doi.org/10.1073/pnas.2017047118

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Fig. 2. Formation of streams and vortices at different surface densities. (A) Structures forming at different surface densities ρ of filaments. (Scale bars, 20 μm.)
(B) Schematic representations of seeds composed of a few filaments, of an elongated stream composed of several filaments, and of a vortex. (C–E) Close-up of
different structures forming. Seeds can merge together (C, ρ ∼0.4 filaments/μm2), and the merging of seeds can result in stream formation (D, ρ ∼0.83 fil-
aments/μm2) or vortex formation (E, ρ ∼0.64 filaments/μm2). Green arrows indicate the local direction of motion. (F) Filaments and seeds stop against a
transient obstacle for about 2 min, and additional filaments with the same orientation further accumulate in place. As the obstacle glides away, a thicker
stream has formed composed of filaments moving in the same direction (Scale bars in C–F, 10 μm.) (G) Snapshots of the system at ρ ∼0.83 filaments/μm2 for
different times and corresponding density histogram showing cluster formation over time. A normalized density of one indicates the isotropic phase. (Scale
bar, 10 μm.) (H) Histogram of the local radius of curvature R of streams (solid lines) and vortices (dashed lines with circles) at different densities. The Inset
shows a schematic of the radius of curvature for a vortex.

such polar bias because of stopping, structures are composed of
polarly moving filaments (see Fig. 4).

Emergence of Polar Order at High Filament Densities. In order to
reach even higher surface densities, another set of experiments is
performed in which actin is added directly together with ATP
and progressively sedimented on the motor-coated SLB by 0.2%
methylcellulose. This allows for experiments with higher actin
bulk concentration (up to 1 μM). By these means, filaments
sediment, move, and align progressively over time, allowing
higher surface concentrations (higher than ρ = 5 filaments/μm2,
corresponding to ∼10% area coverage). Similar to lower densi-
ties, thick streams form, and filaments are still motile within
them. Because of the higher density, streams are now more
persistent and can, after formation, act as an obstacle for fila-
ments traveling in the transverse direction, which then start
stopping upon collision because of the repulsion-dominated in-
teraction. Additional filaments stopping in the same position
lead to a local accumulation of filaments, which hinder each
other’s transverse motion. Local bends in streams are particu-
larly efficient in trapping filaments, acting as a wedge-shaped
trap (49).

The accumulation of filaments in areas where the motility is
low because of an obstacle causes the depletion of isolated freely
moving filaments, which end up either flowing in a stream or
being trapped. This results in big portions of space being de-
pleted from filaments (Fig. 3A and Movie S7). Where filaments
have been trapped by a perpendicularly moving stream, a local
mismatch in the orientation of filaments is present, which re-
sembles that of a classical +1/2 defect (50, 51) (Fig. 3B). As
filaments cannot crawl over each other, the conformation of a
+1/2 defect acts as a “nematic wedge,” which stops filaments and
traps them, as both their forward and lateral motion is pre-
vented, causing an up-concentration of filaments in the core of

the defect. Defects of an opposite charge (−1/2 charge defects)
are also present in approximately the same quantity as positively
charged defects. However, differently from the case of positive
defects, the conformation resulting in −1/2 defects does not
enforce density up-concentration as they do not trap filaments
(Fig. 3B and Movie S8) and the core of the defect is devoid of
filaments (52).

Unlike defects in a two-dimensional nematic system (50, 51,
53), defects in different regions of space do not interact or
merge, as they are not part of a same nematic layer that fully
covers the surface. They instead independently form in locally
dense regions, separated from one another by the low-density
phase and thus can also independently dissolve when filaments
are able to leave the region (Movie S7).

Negative defects eventually dissolve as filaments glide away
along the sides of the defect. Positive wedge-like defects also
eventually dissolve. However, they can still affect the further
behavior of this system. Once the stream previously acting as an
obstacle flows away, the wedge-like obstacle disappears, and
filaments are free to move again. Looking at the intensity profile
along the defect axis over time, one can indeed notice both a first
phase of accumulation of filaments in the presence of a barrier
and a second one in which the defect coherently moves as the
barrier vanishes (Fig. 3 C and D). Since only filaments oriented
in one direction had been trapped in the nematic wedge, as it
dissolves it gives rise to a comet-like stream of filaments, a thick
polar stream that moves persistently. As they arise from a +1/2
defect, filaments inside such a stream conserve a comet-looking
alignment pattern. These streams move and can merge into
other streams or get stuck again if they meet further obstacles.
Thus, nematic wedges act as a polarity sorting mechanism,
whose origin is in the steric interaction between individual fila-
ments, enforcing the tendency to be trapped in convex structures
(5, 49, 54, 55), and in the topological structure of +1/2 defect, in

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Formation of nematic wedges at high filaments’ density. (A) Structures forming at 1 μm bulk actin. (B) Close-up of locally forming +1/2 (Left) and −1/2
Fig. 3.
(Right) topological defects. The local alignment of filaments is also shown, with a mismatch present in the core of the defect. Positive defects up-concentrate
filaments, whereas negative ones are empty. (C) Formation of a comet-like stream emerging from a +1/2 defect. The intensity profile along the dashed line is
also plotted, showing the accumulation and then motion of the comet. Gray profile is the profile at the initial time as a reference. (D) Scheme of the nematic
wedge-based polarity sorting. If a wedge is formed by a preexisting stream (orange), filaments (blue) get trapped inside it unable to escape and accumulate,
forming locally a +1/2 defect. Filaments with opposite polarity (green) are able to escape. Thus, the polarity of filaments is locally sorted. When the obstacle is
removed, filaments are now free to move again and form a comet-like stream. (Scale bars in A–C, 10 μm.)

the case of short actin filaments, which favors accumulation. As a
consequence, streams forming in the high-density case are thick
(more than 10 μm) and strongly polar, with most filaments
traveling in the same direction (Fig. 4C).

Polarity of Observed Patterns. At all densities, streams and vortices
appear to be mostly composed of filaments moving in the same
direction. To investigate the polar nature of the filaments’ di-
rection of motion, we perform FRAP experiments in which an
area of the sample is bleached and then observe the recovery of
the fluorescence intensity (Fig. 4 A–C and Movies S9 and S10).
When bleaching an area inside a stream, both at intermediate
and high density, a recovery starting only from one side can be
observed, indicating the polar motion of the filaments (56). We
monitor this by measuring the asymmetry of the fluorescence
recovery P = |IL−IR|
IL+IR , where IR and IL are the mean fluorescent
intensities on opposite sides of the bleached area (Fig. 4D) av-
eraged over the first 25 s after bleaching. Only one side of the
area is expected to recover if the motion of the filaments is in a
uniform direction (P = 1), whereas a symmetric recovery would
be the result if the motion within the structures is nematic (P ∼
0). At all tested densities, we find values of P above zero.
Streams are, indeed, found to be polar, despite the nematic
nature of the binary interactions (Fig. 4D). At lower densities (up
to ρ ∼0.83 filaments/μm2), we measured P ∼0.6 for streams, in-
dicating polar motion of the filaments within them. The forma-
tion of parallel streams flowing in opposite directions next to
each other can also be observed, leading to symmetric recovery
(P ∼0) of the intensity if both parallel streams are included in the
bleached region and to higher values of P otherwise. Vortices
appear to be composed of filaments that mostly move coher-
ently, even at lower densities (P ∼0.6 to 0.8). As the density is
increased, streams too become noticeably composed of fila-
ments, mostly moving in the same direction. Streams emerging
from +1/2 defects also clearly move in a single direction with
elevated values of P (P ∼0.8 to 0.9).

We attribute the fact that polar structures arise from a nematic
introduced by the
microscopic interaction to the polar bias,
repulsion-dominated interaction, in particular to the stopping of
filaments against transient obstacles. While alignment is nematic
on the individual filaments’ scale, steric repulsion favors accu-
mulation of filaments oriented in the same direction, thus polar
sorting them (Figs. 1D and 2F). We note that the nematic wedge
mechanism we described is particularly effective in enforcing
local polar order by trapping filaments according to their
orientation (Fig. 3).

Discussion
In summary, our experimental system shows that semiflexible
filaments with hard volume exclusion interaction and alignment
self-organize into locally polar phases. The observed structures
arise from the interplay of filaments’ flexibility, thermal fluctu-
ations, and a microscopic interaction preventing filaments from
crawling over each other. Here, the membrane does not act as an
immobile momentum sink during collision, which is usually as-
sumed in the simulation of self-propelled rods and motivated in
most experimental setups (5, 20), but rather, momentum is
transferred to the lipid bilayer (31). This effectively limits the
filaments’ ability to push upon collisions and leads to the steri-
cally enforced stopping of filaments upon collisions and succes-
sive alignment. Stopping also plays a role in the accumulation of
filaments in the presence of obstacles and, thus, in their polarity
sorting. This allows breaking the microscopic nematic symmetry
and is at the base of the formation of polar streams and vortices,
which are different from those obser