# Locomotion of Active Polymerlike Worms in Porous Media

R. Sinaasappel  Van der Waals-Zeeman Institute, Institute of Physics,
University of Amsterdam, 1098XH Amsterdam, The Netherlands.  M. Fazelzadeh
Institute for Theoretical Physics, University of Amsterdam, Science Park 904,
1098XH Amsterdam, The Netherlands.  T. Hooijschuur  Van der Waals-Zeeman
Institute, Institute of Physics, University of Amsterdam, 1098XH Amsterdam,
The Netherlands.  Institute for Theoretical Physics, University of Amsterdam,
Science Park 904, 1098XH Amsterdam, The Netherlands.  S. Jabbari-Farouji
<EMAIL_ADDRESS>Institute for Theoretical Physics, University of
Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands.  A. Deblais
<EMAIL_ADDRESS>Van der Waals-Zeeman Institute, Institute of Physics,
University of Amsterdam, 1098XH Amsterdam, The Netherlands.

###### Abstract

We investigate the locomotion of thin, living T. Tubifex worms, which display
active polymerlike behavior, within quasi-2D arrays of pillars with different
spatial arrangements and densities. These active worms spread in crowded
environments, with a dynamics dependent on both the concentration and
arrangement of obstacles. In contrast to passive polymers, our results reveal
that in disordered configurations, increasing the pillar density enhances the
long-time diffusion of our active polymer-like worms, while we observe the
opposite trend in ordered pillar arrays. We found that in disordered media,
living worms reptate through available curvilinear tubes, whereas they become
trapped within pores of ordered media. Intriguingly, we show that reducing the
worm’s activity significantly boosts its spread, enabling passive sorting of
worms by activity level. Our experimental observations are corroborated
through simulations of the tangentially-driven polymer model.

Polymer physics, Active polymers, porous media

###### pacs:

Valid PACS appear here

Biological organisms exhibit a diverse array of features to optimize their
survival strategies and effectively navigate through their disordered natural
habitats [1, 2, 3, 4, 5, 6, 7, 8]. From actin filaments in the cytoskeleton to
larger-scale examples like cyanobacteria in lakes or earthworms in soil,
active filaments are observed moving through complex and crowded environments.
However, despite their abundance and importance, the mechanisms by which
motile filaments navigate through porous media remain poorly understood [9,
10, 11].

Few, mostly theory-based modeling studies have been performed on the dynamics
of active filaments in complex environments [12, 13, 14, 15, 16, 17]. These
investigations have predominantly focused on active Brownian and tangentially
driven polymer models, delving into the interplay of length, flexibility, and
activity on polymer dynamics, both within porous media [18, 15, 16, 17] and
confined spaces [19]. In the case of active stiff or semi-flexible polymers,
they move through the porous medium smoothly with motion that closely
resembles reptation, a concept originally elucidated by de Gennes for passive
a polymer performing wormlike displacements in arrays of fixed obstacles [20].
In the case of very flexible active polymers however, they will bundle up
inside the pores, causing hopping-trapping dynamics where the polymers are
stuck in the pores and only rarely hop between pores [18, 17]. In these
investigations, the effect of porous medium density and particularly the
geometry of the pattern on the dynamics of active polymers has received little
attention. Additionally, accurate modeling of migration for active polymers in
porous media has been limited because of a lack of suitable experimental model
systems to investigate these questions.

Figure 1: Trajectories of the center of mass an active polymerlike worm in
$L^{2}$ = 44$\times$44 cm2 2D array of pillars with a diameter of 5 mm and
surface fraction $\phi$ = 0.4 (only a part of medium is shown) in a (a) random
packing and (b) square lattice for a duration of 185 s; scale bar corresponds
to 15 mm. Close-ups show the distinct conformations of a worm arising in the
two different obstacle arrangements. (c),(d) Center-of-mass trajectories and
representative polymer conformations of the simulated tangentially driven
active filaments in the same geometries as in (a),(b). Figure 2: (a) Effective
persistence length $l_{p}^{eff}/L_{c}$ of living worms versus tangentially
driven active polymer as a function of $\phi$. (b) Time auto-correlation
function of the end-to-end vector, determining the reorientational relaxation
time $\tau_{e}^{0}$ of the worm (continuous line, $\tau_{e,0}\sim 40s$) and of
the tangentially driven polymer (dotted lines) in free space.

In this study, we use T. Tubifex worms, which exhibit behavior resembling that
of active polymers [21, 22, 23, 24], to investigate locomotion within quasi-2D
porous media consisting of cylindrical pillars. Comparisons are made with a
computational model of tangentially driven active filaments, commonly used to
represent self-propelled biopolymers [18, 25, 17]. Our results reveal the
dynamics of the worms depend on the surface fraction $\phi$ occupied by the
pillars and the degree of order in obstacle arrangements. In disordered pillar
arrays, the long-time diffusion coefficient $D_{l}$ increases significantly
with $\phi$, whereas in square-lattice arrangements, the opposite trend is
observed. This transport enhancement in disordered media is attributed to the
worms’ ability to reptate through accessible curvilinear tubes as formed by
randomly positioned pillars, contrasting with their entrapment within voids in
ordered configurations. Surprisingly, reducing the worm’s activity boosts its
spread in disordered media due to an increase in its effective persistence
length, which facilitates persistent trajectories within available tubes.
Furthermore, our simulations confirm that modeling the worms as tangentially
driven active filaments accurately accounts for the long-time dynamics
observed in T. Tubifex worms.

When deposited on a free surface, the wiggling motions of the T. Tubifex allow
the worm to crawl on the surface, giving rise to a diffusive motion. The
crawling motion is effectively two-dimensional (2D), since the worms are
denser than water and, therefore, always located at or close to the bottom
part of the geometry.

In our experiments, we introduced a single T. Tubifex worm in arrays of
pillars with radius $R_{p}$ = 2.5 mm on a two-dimensional square surface
44$\times$44 cm2 of the same material which is immersed in a thermostated
water volume and we tracked its motion in real time by recording videos for a
duration of 2 hours. We investigate locomotion of worms at room temperature
$T$ = 21°C in two distinct pillar arrangements. In the first one, the worms
are dispersed in a periodic (square lattice) structure. In the second
arrangement, we create a disordered array of pillars by randomly distributing
$N$ static pillars with the geometric constraint that the pillars do not
overlap. The minimum distance between any two pillars is set to the
characteristic width of a worm (approx. 0.5 mm) to ensure that worms can pass
through; see Sec. I and Fig. S1 and Fig. S2 in the Supplementary Materials
[26] for details. We varied the pillar surface fraction in the range
$[0.1,0.6]$. All the reported conformational and dynamical features of worms
at $T$ = 21°C have been obtained by averaging at least 30 worm trajectories
selected with a constant contour length $L_{c}$ = 23 $\pm$ 8 mm; we checked
that the contour length does not play a significant role within this range
(Sup. Fig. S3, [26]).

Figure 3: Mean square displacements (MSD) normalized by contour lengths
$l_{c}$ as a function of time normalized by reorientational time of end-to-end
vector $\tau_{e,0}$ for living worms in water at $T$ = 21°C in disordered (a)
and ordered (b) obstacle arrangements. In the disordered pattern (a), the MSD
increases faster as surface fraction $\phi$ rises, while the reverse trend is
observed in the ordered pattern (b). Dashed lines represent simulations of the
tangentially driven model for $\phi$ = 0 (free) and 0.4. (c) The slopes
$\alpha$ of the MSD curves in the intermediate regime for both living worms
(filled) and simulations of active polymers (open) in both geometries. (d) The
long-time diffusion coefficient $D_{L}$, normalized with the diffusion
coefficient in free space $D_{L}^{0}$, as a function of $\phi$. Lines guide
the eyes. The inset shows the linear relationship between $D_{l}$ and
$l_{p}^{eff}/L_{c}$. Figure 4: (a) Example of a worm trajectory ($\phi=40\%$)
as it reptates within an effective tube made by the disordered positioning of
pillars. (b) Worms are defined as caged when their instantaneous speed drops
below a cutoff of $0.5~{}mm/s$. Between caging events, worms crawl effective
tubes of length $L_{tube}$. (c) and (d) Distribution of tube lengths for
disordered (c) and ordered (d) media. Longer tube lengths are observed as
$\phi$ increases in the disordered case, while the maximum tube length
decreases for higher $\phi$ in the ordered medium. The opposite trend is
observed for the distribution of trapping times in the disordered (e), and
ordered media (f). Open symbols are results from simulations.

In Figure 1, we show typical trajectories of the center-of-mass of a worm
moving in a disordered (a) and ordered (b) pillar arrays. In both geometries,
as time progresses, the worm is able to navigate through the obstacles but the
dynamics with which they do so is found to be dependent on the lattice order
and density. To investigate the similarities with active polymers, we
conducted simulations using the tangentially-driven polymer model, as
described in [27, 17] (see also Sec. II in Sup. Mat. [26]). The motion of each
monomer follows overdamped Langevin dynamics, including the active force of
amplitude $f^{a}$ per monomer along the tangent of the backbone of the polymer
and the bending stiffness $\kappa$ between neighboring bonds, which represents
the inherent flexibility of the active polymer. The key parameters of the
model thus boil down to ($f^{a}$, $\kappa$). To compare the tangentially
driven polymers to the worms we measure their tangent-vector orientational
correlations and average over time (Sup. Fig. S4& S5, [26]). From this we
extract the effective persistence length $l_{p}^{eff}$, which correlates to
$\kappa$. The effective persistence length $l_{p}^{eff}$ is shown by bold
symbols in Fig. 2(a) as a function of $\phi$ in ordered and disordered media,
which shows an increase (decrease) with $\phi$ for worms in ordered and
disordered media. Then, we adjust the parameter $\kappa$ in our simulations to
match the observed $l_{p}^{eff}$, see the open symbols in Fig. 2(a).

The active force $f^{a}$ is selected to ensure it dominates at the level of
the whole polymer while still allowing thermal fluctuations to affect
individual monomers. To compare experiments and simulations, we rescale the
time by the reorientational relaxation time of the worm, determined from the
$(1/e)$-decay time of the autocorrelation function of the end-to-end vector,
see Fig. 2(b). For tangentially driven polymers in free space with
sufficiently large active force ($f^{a}>0.01$), both longtime diffusion,
dominated by activity, and the orientational relaxation time of the end-to-end
vector ($\tau_{e}$) scale linearly with active force per monomer ($f^{a}$).
Therefore, rescaling the time by $\tau_{e}$ eliminates the explicit dependence
on the active force, making the precise value of $f^{a}$ less critical [28,
29].

We examine the effect of the geometry of the obstacle arrays and the
concentration of the pillars on the motion of a self-locomoting worm by
measuring the mean squared displacement (MSD) as shown in Figure 3(a) and (b).
The time is non-dimensionalized by normalizing with the worm’s reorientational
relaxation time in a free environment $\tau_{e,0}\approx$ 40 s. For short time
scales ($\tau/\tau_{e,0}\leq 0.02$), the worms have not yet encountered the
pillars, resulting in only slight superdiffusive behavior with a diffusion
exponent of 1.2. After this typical time scale, our living worms interact with
the pillars with a dynamics that depends on the degree of environmental order.
In the disordered case, we observe an almost ballistic regime as shown by
$\alpha\sim 1.8$ in Fig. 3(a), which we found to be almost independent of the
surface fraction of the pillar $\phi$ [Fig. 3(c)]. At long times
$\tau/\tau_{e,0}\gg 1$, each MSD curve exhibits a diffusive regime where
$MSD=4D_{l}t$, with a slope that surprisingly increases with $\phi$. For the
ordered media [Fig. 3(b)], the opposite trend is observed: We observe a weak
superdiffusive regime and almost a direct crossover from short-time dynamics
to the diffusive regime with a slope that decreases with increasing $\phi$. As
a result, the presence of an intermediate regime of super-diffusion depends on
the arrangement of obstacles, whether they are ordered or not.

The spatial arrangement and density of obstacles influence the long-time
diffusion coefficient $D_{L}$ of the active worms. As shown in Fig. 3(d), the
two geometries exhibit opposite effects: in disordered media, $D_{l}$
increases with $\phi$, while it decreases in ordered media, obeying the
intuitive expectation that greater crowding slows diffusion. Interestingly,
the time scales $\tau\sim\tau_{e,0}$, at which we observe the diffusive
regime, do not appear to depend on the obstacle surface fraction $\phi$.
Moreover, extracting $\tau_{e}(\phi)$ from the orientational-time
autocorrelation of the end-to-end vector (see Sup. Fig. S6 in [26]) confirms
that $\tau_{e}$ is independent of $\phi$, consistent with simulation results
for semiflexible active polymers in periodic obstacle arrays [17].

Examining the trajectories of the worm’s center of mass in ordered and
disordered pillar arrays, shown in Figs. 1(a),(b), suggests that the
contrasting trends in diffusion originate from distinct locomotion strategies.
In disordered media, worms and active polymers, see Figs. 1(a),(c), move
through effective tubes formed by randomly positioned pillars. They switch
between these tubes after brief trapping periods, similar to the reptation of
thermal polymers in crowded disordered environments [20, 30, 31, 32], where
each pore acts as an entropic trap. Between runs, worms become trapped in the
pores of the geometries by coiling up, as depicted in Fig. 1(b). Conversely,
in the ordered structure, worms appear more flexible, as shown by the lower
persistence length in Fig. 3(e). They spend more time within the pores and
occasionally hop from one pore to another, with elongated trajectories within
narrow throat regions.

This is reflected in the distributions of the curvilinear tube length within
which worms and active polymers reptate and their trapping times. Trapping
events are determined as the time intervals in which the instantaneous speed
of the center of mass drops below $v=0.5$ mm/s, and the trajectory of the
center of mass is localized in space; see Figs. 4(a),(b). We define a tube
length $L_{tube}$ as the distance a worm travels between trapping events,
corresponding to ballistic-like motion within curvilinear tubes formed by the
adjacent obstacles. Figs. 4(c),(d) shows the probability distribution function
of the tube run length $P(L_{tube})$ as the worms reptate through available
curvilinear tubes. In disordered media, higher pillar densities correlate with
longer run lengths within reptation tubes, as evidenced by the larger
$P(L_{tube})$ in the tail of the distribution in Fig. 4(c). In contrast, in
ordered media, increasing $\phi$ results in shorter available tube lengths, as
reflected by the decrease of $P(L_{tube})$ in the tail with $\phi$ in Fig.
4(d).

Next, we examine the distribution of trapping times $\tau_{trap}$, defined as
the duration of trapping events. Figs. 4(e),(f) show $P(\tau_{trap})$ as a
function of $\tau_{trap}$ for disordered and ordered media, respectively. In
disordered media, the tails of the distribution decrease with increasing
$\phi$, consistent with our observation that worms spend more time in tubes in
this configuration [Fig. 4(e)]. However, in ordered media, the distribution
remains almost unchanged with increasing $\phi$ [Fig. 4(f)]. Our tangentially
driven model (open symbols) captures the general shape of the distributions
but not the exact dependence on $\phi$. These findings invite further scrutiny
for both modeling and experiments. Note that the ordered geometry at
$\phi=0.5$ is a hexagonal lattice with a few defects rather than e square
lattice, leading to a non-monodisperse void size distribution with a small
log-normal contribution. For clarity, these points are omitted from the figure
but are shown in Sup. Fig. S7.

The contrasting trends in both the dependence of the long-time diffusion
coefficient and the distribution of the run length within tubes on the pillar
packing fraction $\phi$ for ordered and disordered media originate from
changes in the effective persistence length of the worms with increasing
$\phi$. The active contribution to long-time diffusion ($D_{l}^{a}$) is
dominant, and theoretical calculations for tangentially-driven active chains
predict that it scales as $D_{l}^{a}={f^{a}}^{2}\langle
R_{e}^{2}\rangle\tau_{e}/(\gamma^{2}L_{c}^{2})$, irrespective of $\phi$ [17],
also valid in free space [28, 29] (with the average end-to-end distance
$\langle R_{e}^{2}\rangle$ and friction coefficient $\gamma$). Additionally,
$\tau_{e}\sim L_{c}/f^{a}$ leads to $D_{l}^{a}\sim f^{a}\langle
R_{e}^{2}\rangle/L_{c}$. For the worms, the contour length $L_{c}$ and the
friction coefficient $\gamma$ do not change. Experimentally measured values of
$\tau_{e}$ remain constant regardless of variations in $\phi$. Assuming the
worm’s activity level $f^{a}$ does not depend on the geometry, changes in
$\langle R_{e}^{2}\rangle$ in porous media determine the long-time diffusion.
While accurately measuring $\langle R_{e}\rangle$ experimentally is
challenging, we can find a value if we assume the worm-like-chain model for
our semiflexible polymer-like worms; $\langle
R_{e}^{2}\rangle=2l_{p}^{eff}L_{c}$ when $l_{p}^{eff}\ll L_{c}$. Thus, we
expect $D_{l}\sim l_{p}^{eff}$, as confirmed in the inset of Fig. 3(d).

However, it is worth noting that the tangentially-driven polymer model does
not fully capture detailed dynamics, such as the exponent of anomalous
diffusion in intermediate regime of MSD as depicted in Fig. 3(c). To achieve
complete agreement between experiments and theory, alternative theoretical
models are necessary that allow for heterogeneous or time-dependent active
force distributions or transversal motion modes [33, 34, 35, 36].

Figure 5: (a) MSDs of T. Tubifex worms at high ($T=30^{\circ}$C, red dots),
low ($T=5^{\circ}$C, blue dots) and intermediate ($T=21^{\circ}$C, silver
dots) activity levels. Time is normalized by the respective reorientational
timescales ($\tau_{e}^{5\celsius}=78$ s, $\tau_{e}^{21\celsius}=41$ s,
$\tau_{e}^{30\celsius}=36$ s), which are temperature dependent (Sup. Fig. S9
[26]). Dashed lines show tangentially driven polymer simulations with matching
persistence length and $f^{a}=0.1$. (b) MSDs of worms at high and low
temperatures in disordered porous media. Time is normalized by $\tau_{e}^{T}$.
At lower temperatures, the effective persistence length increases, enhancing
worm spread compared to high temperatures ($T=30^{\circ}$C, red line). Dashed
lines correspond to simulation results adjusted for $\kappa$.

Finally, we examine the impact of activity on worm transport, adjustable via
ambient water temperature [22]. The average persistence length of worms
remains unchanged with temperature [23], whereas their long-time diffusion
coefficient ($D_{l}$) increases, and their reorientational time ($\tau_{e}$)
decreases with temperature [22]. This behavior aligns with tangentially-driven
active polymers, where $D_{l}$ scales linearly with activity force ($f^{a}$)
and $\tau_{e}$ scales as $L_{c}/f^{a}$ [29, 28]. Thus, we expect the mean
squared displacement (MSD) versus time, scaled by $\tau_{e}^{T}$, to be
temperature-independent. Fig. 5(a) shows the MSD for worms at three activity
levels (low, T = $5\celsius$, intermediate, T= $21\celsius$, and high, T =
$30\celsius$). While higher temperatures result in larger MSDs, normalizing by
$\tau_{e}^{T}$ largely mitigates this effect (See Sup. Fig. S8, [26] for the
same data in lab-units).

In disordered environments, however, the MSD versus rescaled time
($\tau/\tau_{e}^{T}$) strongly depends on activity level, as shown in Fig.
5(b). Surprisingly, worms with lower activity (T = $5^{\circ}$C) exhibit
faster long-time diffusion than those with higher activity (T =
$30^{\circ}$C), despite the increase in $\tau_{e}$ at lower temperatures. This
suggests that the contrasting trend is not due to increased worm activity in
the porous medium at lower temperatures. Interestingly, the effective
persistence length of worms in disordered media, averaged over the
experimental timescale (Sup. Fig. S9 [26]), increases at lower temperatures
from $l_{p}^{eff}$ = 0.12 to 0.6.

To rationalize these experimental findings, simulations are conducted in the
same geometry. The active force is kept constant as in previous experiments
and the bending stiffness values are selected to match the effective
persistence length measured for worms ($30\celsius:\kappa=5.5$,
$5\celsius:\kappa=40$, dashed lines). We observe good agreement between worm
behavior and simulations of tangentially driven polymers at large timescales
for low-activity worms. However, at high temperatures, the tangentially driven
model no longer adequately describes worm dynamics. This suggests that worms
under low-temperature conditions and confined by pillars, primarily exhibit
peristaltic-like motion. Conversely, at higher activity levels, worms exhibit
more transversal modes of motion, indicating the necessity for a more refined
model to accurately capture their behavior.

In conclusion, our study highlights the crucial role of environmental order,
porosity, and activity in shaping the dynamics of active flexible agents.
Unlike previous work on active stiff and semi-flexible polymers [5, 12], we
demonstrate that pore morphology significantly influences the large-scale
spreading of living worms. These findings highlight the importance of pore
geometry in modulating transport phenomena within porous environments,
suggesting potential for manipulation through lattice patterning. Moreover,
our observations suggest that strategically distributing active forces along a
driven polymer could enhance active transport, analogous to the behaviors
observed in our living worms.

## References

  * Dorgan  _et al._ [2013] K. M. Dorgan, C. J. Law, and G. W. Rouse, Meandering worms: mechanics of undulatory burrowing in muds, Proceedings of the Royal Society B: Biological Sciences 280, 20122948 (2013).
  * Bechinger  _et al._ [2016] C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Reviews of Modern Physics 88, 045006 (2016).
  * Sosa-Hernández  _et al._ [2017] J. E. Sosa-Hernández, M. Santillán, and J. Santana-Solano, Motility of escherichia coli in a quasi-two-dimensional porous medium, Phys. Rev. E 95, 032404 (2017).
  * Makarchuk  _et al._ [2019] S. Makarchuk, V. C. Braz, N. A. M. AraÃºjo, L. Ciric, and G. Volpe, Enhanced propagation of motile bacteria on surfaces due to forward scattering, Nature Communications 10, 4110 (2019).
  * Bhattacharjee and Datta [2019] T. Bhattacharjee and S. S. Datta, Bacterial hopping and trapping in porous media, Nature Communications 10, 2075 (2019).
  * Kudrolli and Ramirez [2019] A. Kudrolli and B. Ramirez, Burrowing dynamics of aquatic worms in soft sediments, Proceedings of the National Academy of Sciences 116, 25569 (2019), publisher: Proceedings of the National Academy of Sciences.
  * Dehkharghani  _et al._ [2023] A. Dehkharghani, N. Waisbord, and J. S. Guasto, Self-transport of swimming bacteria is impaired by porous microstructure, Communications Physics 6, 1 (2023).
  * Juarez  _et al._ [2010] G. Juarez, K. Lu, J. Sznitman, and P. E. Arratia, Motility of small nematodes in wet granular media, Europhysics Letters 92, 44002 (2010).
  * Lux  _et al._ [2001] R. Lux, J. N. Miller, N.-H. Park, and W. Shi, Motility and Chemotaxis in Tissue Penetration of Oral Epithelial Cell Layers by Treponema denticola, Infection and Immunity 69, 6276 (2001).
  * Thornlow  _et al._ [2015] D. N. Thornlow, E. L. Brackett, J. M. Gigas, N. Van Dessel, and N. S. Forbes, Persistent enhancement of bacterial motility increases tumor penetration, Biotechnology and Bioengineering 112, 2397 (2015).
  * Toley and Forbes [2012] B. J. Toley and N. S. Forbes, Motility is critical for effective distribution and accumulation of bacteria in tumor tissue, Integrative Biology: Quantitative Biosciences from Nano to Macro 4, 165 (2012).
  * Theeyancheri  _et al._ [2023] L. Theeyancheri, S. Chaki, T. Bhattacharjee, and R. Chakrabarti, Active dynamics of linear chains and rings in porous media, The Journal of Chemical Physics 159, 014902 (2023).
  * Majmudar  _et al._ [2012] T. Majmudar, E. E. Keaveny, J. Zhang, and M. J. Shelley, Experiments and theory of undulatory locomotion in a simple structured medium, Journal of The Royal Society Interface 9, 1809 (2012), publisher: Royal Society.
  * Mokhtari and Zippelius [2019a] Z. Mokhtari and A. Zippelius, Dynamics of active filament in porous media, Phys. Rev. Lett. 123, 028001 (2019a).
  * Kurzthaler  _et al._ [2021] C. Kurzthaler, S. Mandal, T. Bhattacharjee, H. LÃ¶wen, S. S. Datta, and H. A. Stone, A geometric criterion for the optimal spreading of active polymers in porous media, Nature Communications 12, 7088 (2021).
  * Tejedor  _et al._ [2023] A. R. Tejedor, R. Carracedo, and J. Ramírez, Molecular dynamics simulations of active entangled polymers reptating through a passive mesh, Polymer , 125677 (2023).
  * Fazelzadeh  _et al._ [2023a] M. Fazelzadeh, Q. Di, E. Irani, Z. Mokhtari, and S. Jabbari-Farouji, Active motion of tangentially driven polymers in periodic array of obstacles, The Journal of Chemical Physics 159, 224903 (2023a).
  * Mokhtari and Zippelius [2019b] Z. Mokhtari and A. Zippelius, Dynamics of active filaments in porous media, Physical Review Letters 123, 028001 (2019b).
  * MartÃn-Roca  _et al._ [2024] J. MartÃn-Roca, E. Locatelli, V. Bianco, P. Malgaretti, and C. Valeriani, Tangentially active polymers in cylindrical channels (2024), arXiv:2405.02192 [cond-mat.soft] .
  * de Gennes [1971] P. G. de Gennes, Reptation of a polymer chain in the presence of fixed obstacles, The Journal of Chemical Physics 55, 572 (1971), publisher: American Institute of Physics.
  * Deblais  _et al._ [2020a] A. Deblais, S. Woutersen, and D. Bonn, Rheology of entangled active polymer-like t. tubifex worms, Physical Review Letters 124, 188002 (2020a).
  * Deblais  _et al._ [2020b] A. Deblais, A. Maggs, D. Bonn, and S. Woutersen, Phase separation by entanglement of active polymerlike worms, Physical Review Letters 124, 208006 (2020b).
  * Heeremans  _et al._ [2022] T. Heeremans, A. Deblais, D. Bonn, and S. Woutersen, Chromatographic separation of active polymer-like worm mixtures by contour length and activity, Science Advances 8, eabj7918 (2022).
  * Deblais  _et al._ [2023] A. Deblais, K. R. Prathyusha, R. Sinaasappel, H. Tuazon, I. Tiwari, V. P. Patil, and M. S. Bhamla, Worm blobs as entangled living polymers: from topological active matter to flexible soft robot collectives, Soft Matter 19, 7057 (2023).
  * Winkler and Gompper [2020] R. G. Winkler and G. Gompper, The physics of active polymers and filaments, The Journal of Chemical Physics 153, 040901 (2020).
  * [26] See Supplemental Material [url] for more details on the experimental methods, set-up and on the simulations of the tangentially-driven polymer model. It includes the references [21, 27, 37, 38, 17, 29, 30].
  * Isele-Holder  _et al._ [2015] R. E. Isele-Holder, J. Elgeti, and G. Gompper, Self-propelled worm-like filaments: spontaneous spiral formation, structure, and dynamics, Soft Matter 11, 7181 (2015).
  * Fazelzadeh  _et al._ [2023b] M. Fazelzadeh, E. Irani, Z. Mokhtari, and S. Jabbari-Farouji, Effects of inertia on conformation and dynamics of tangentially driven active filaments, Phys. Rev. E 108, 024606 (2023b).
  * Bianco  _et al._ [2018] V. Bianco, E. Locatelli, and P. Malgaretti, Globulelike conformation and enhanced diffusion of active polymers, Phys. Rev. Lett. 121, 217802 (2018).
  * Doi and Edwards [1986] M. Doi and S. F. Edwards,  _The Theory of Polymer Dynamics_ (Clarendon Press, Oxford, 1986).
  * Muthukumar and Baumgärtner [1989a] M. Muthukumar and A. Baumgärtner, Diffusion of a polymer chain in random media, Macromolecules 22, 1941 (1989a).
  * Muthukumar and Baumgärtner [1989b] M. Muthukumar and A. Baumgärtner, Effects of entropic barriers on polymer dynamics, Macromolecules 22, 1937 (1989b).
  * Vatin  _et al._ [2024] M. Vatin, S. Kundu, and E. Locatelli, Conformation and dynamics of partially active linear polymers, Soft Matter 20, 1892 (2024).
  * Anand and Singh [2018] S. K. Anand and S. P. Singh, Structure and dynamics of a self-propelled semiflexible filament, Physical Review E 98, 042501 (2018), publisher: American Physical Society.
  * Anand and Singh [2020] S. K. Anand and S. P. Singh, Conformation and dynamics of a self-avoiding active flexible polymer, Physical Review E 101, 030501 (2020), publisher: American Physical Society.
  * Prathyusha  _et al._ [2022] K. Prathyusha, F. Ziebert, and R. Golestanian, Emergent conformational properties of end-tailored transversely propelling polymers, Soft Matter 18, 2928 (2022).
  * Weeks  _et al._ [1971] J. D. Weeks, D. Chandler, and H. C. Andersen, Role of repulsive forces in determining the equilibrium structure of simple liquids, The Journal of Chemical Physics 54, 5237 (1971).
  * Binder  _et al._ [2020] K. Binder, S. A. Egorov, A. Milchev, and A. Nikoubashman, Understanding the properties of liquid-crystalline polymers by computational modeling, Journal of Physics: Materials 3, 032008 (2020).

## Supplementary Materials for
“Locomotion of Active Polymerlike Worms in Porous Media”

This Supplementary Material provides additional information on the
experimental setup and methodologies employed in this study and on the
tangentially driven polymer model utilized for our analysis.

##  I EXPERIMENTS

###  I.1 Experimental set-up

In our experiments, we placed a single T. Tubifex worm into pillar arrays
submerged in a thermostated water volume, tracking their motion in real-time
through 2-hour video recordings. We investigated two geometries: (i) a
periodic crystalline structure and (ii) a disordered geometry with randomly
positioned pillars.

For each pattern, we inserted $N$ static pillars with a radius $R_{p}$ = 2.5
mm on a square two-dimensional surface of the same material, ensuring no
overlap between the pillars. We imposed the condition that the minimum
distance between two pillars is approximately the characteristic width of a
worm ($\sim$ 500 $\mu$m) to allow worm passage. The surface fraction of the
pillar varied from $10\%$ to $60\%$, calculated as
$\phi=N\pi{R_{p}}^{2}/L^{2}$. In the disordered medium, pillar positions were
randomly selected using the numpy.random library in Python. Pillar placement
continued sequentially, retrying if a selected location was closer than 2.5+1
mm to a previously placed pillar, until reaching the desired surface fraction
$\phi$. Supplementary Figure SS1 shows the experimental geometries, with
dimensions $L^{2}$ of 230x230 mm2 for ordered and 440x440 mm2 for disordered
setups. The larger setup involved cutting holes in an acrylic sheet and
placing short acrylic rods with a radius of 2.5 mm; the ordered geometries
were 3D-printed. We observed no significant differences in the behavior of the
worm between the two dimensions.

To quantify the size of the voids in the different geometries, we used
Delaunay triangluation (as implemented in the scipy.spatial library). Delaunay
triangluation divides an area containing a set of points into triangles, such
that the circumcircles of these triangles do not contain any points. It allows
us to find the biggest possible circles that one can draw that do not contain
any of the pillars. The radius of these circles are taken as a good
approximation for the void sizes in the geometries, so it is possible to
compare how the void size distribution changes with the surface fraction of
the pillars. Supplementary Figure SS2 shows the distribution of void radii in
the pillar arrays, following a log-normal distribution. For larger surface
fraction above $\phi=50\%$, we were unable to produce a disordered medium
using the standard method. Instead, we started from an evenly spaced hexagonal
lattice, randomly removing pillars to achieve the desired surface fraction.
Next, the pillars where allowed to diffuse for a while. This resulted in a
(largely) hexagonal lattice with defects, exhibiting a more mono-disperse void
size distribution. Due to this, the worms behaved similarly to those in the
square lattice geometries, therefore they will be referred to as ordered in
subsequent materials.

Figure S1: The ordered and disordered geometries used in the experiments. The
axis are in millimeters and the radius of the pillars is $r$ = 2.5 mm. The
surface fraction occupied by the pillars is indicated in the top left corner
in red. The bottom left corner shows the Delaunay triangulation. Figure S2:
Pore size distribution in the geometries. Distribution of the radii of all
possible biggest circles that do not contain any points, as found through
Delaunay triangulation, as a measure for the size of the voids in the pillar
arrays. The distribution of the void sizes are well fitted with a log-normal
distribution. For $\phi$ = 50% the distribution is fitted by the sum of a
Gaussian and a log-normal distribution.

####  I.1.1 Tracking and mean square displacement (MSD)

In the experiments, worms are positioned atop the geometries in a 15 cm deep
water bath, dimly illuminated from below with an LED panel. The camera
recordings from above capture the experiments (Nikon D5300 equipped with a
macrolens), which are subsequently analyzed using a Python script. From the
images, we extracted the center of mass (CoM) and the contour of the worm, as
shown in Figure SS4.

After tracking the worm’s CoM position ($\textbf{r}=(x,y)$), we compute the
mean square displacement as a function of lag time
$\mathrm{MSD}(\tau)=\langle(r(\tau)-r(0))^{2}\rangle$. Each experiment yields
one MSD curve, which is then averaged to produce the curves reported in the
main paper. Supplementary Figure SS3 displays the MSD curves of all
experiments with the average represented by the black line.

###  I.2 Effect of worm’s contour length, $L_{c}$

In our experiments, worms of varying lengths were used due to experimental
constraints (batch of worms are polydisperse in their contour lengths).
However, we did not observe any significant correlation between the length of
the worms and their behavior. Analysis of the long-time diffusion constant and
MSD curves revealed no discernible correlation with the length of the worm, as
illustrated in Figure S3(a-f), where the color code represents the contour
length of the worm.

Figure S3: MSD curves of all experiments. (a) Long-time diffusion constant
versus the contour length of worms in the disordered, (b) ordered media, and
(c) in free space. (d) MSD curves of worms in free space in lab units. The
color of the lines indicates the contour length of the worms. The average is
indicated by the black line. (e) All MSD curves for the disordered medium. The
color of the lines is mapped according to the color bar in (d). The surface
fraction of the pillars is indicated in the top left corner. (f) All MSD
curves for the ordered medium. The color of the lines is mapped according to
the color bar in (d). The surface fraction of the pillars is indicated in the
top left corner.

###  I.3 Conformation

####  I.3.1 Effect of the geometry and temperature on the persistence length

Figure S4: Determination of the persistence length.

To determine the appropriate bending stiffness $\kappa$ of the semi-flexible
polymers in the simulations, we calculated the effective persistence length
$l_{p}^{eff.}$ from the videos of the worms for different surface fractions
and in ordered and disordered media. Calculating the persistence length
involves defining an effective bond vector for the worm. Firstly, we remove
the background from each image and select a smaller region of interest (ROI)
of the total image where the worm is either 600x600 or 360x360 pix2, depending
on the total image size. Within this selected image region, we identify all
pixels belonging to the individual worm and employ a skeletonization algorithm
to obtain a single-pixel-wide chain of pixels, which forms the initial
polymer-like backbone of the worm. Each pixel can be regarded as a monomer,
with neighboring pixels connected by bond vectors $\textbf{r}_{i,j}$.

However, at the single-pixel level, each neighbor has only 8 possible
directions it can be connected, resulting in discrete bond vectors and sharp
lines in the bond-bond correlation function. Additionally, at high resolution,
each pixel represents a minute scale, rendering the bonds effectively rigid,
leading to a plateau in the bond-bond correlation. To address these issues, we
average every four pixels to create a new monomer $\textbf{r}_{i}^{*}$, and
these are connected by new bond vectors, denoted as $\textbf{b}_{i}$, where
$i$ ranges from 0, the first bond in the chain, to $i=N-1$, the last bond.

Figure S5: Effect of the lattice pattern on the persistence length. (a) Bond-
Bond correlation of the worms in the disordered medium. The persistence length
increases with increasing the packing fraction (b) The effect of the
increasing packing fraction in a ordered arrangement on the persistence
length. The bond-bond correlation curves are averaged in time over around 10
minutes of footage per worm for 5 different worms.

To determine the bond-bond correlation function, we examine the dot product
between a normalized bond vector $\hat{\textbf{b}}_{i}$ at the point $i$ and a
bond vector $s$ that bonds along the chain $\hat{\textbf{b}}_{i+s}$. This dot
product yields an angle $\cos(\theta)$ between the bonds, which decreases as
we move along the contour $s$ until it becomes completely uncorrelated at
$\cos(\theta)=0$. In molecular dynamics simulations of polymers, this decay
follows a decreasing exponential pattern, given by
$\langle\hat{\textbf{b}}_{i+s}\cdot\hat{\textbf{b}}_{i}\rangle=e^{-s/l_{p}}$,
where $l_{p}$ represents the persistence length of the worm. The averaging
$\langle...\rangle$ involves both a time and a sample average, incorporating
different frames of the same worm, as well as averaging over trajectories of
different worms.

To extract the persistence length from the bond-bond correlation function, we
fit the exponential function to the data for the first 5 mm of the worm
contour. Long-range correlations are less accurate because of fewer frames in
which the entire worm is visible, often caused by overlaps or obstructing
pillars. An alternative method is to use the crossing point of
$\langle\hat{\textbf{b}}_{i+s}\cdot\hat{\textbf{b}}_{i}\rangle$ at $1/e$, but
for very stiff worms, this crossing point may not always be reached for high
contour lengths. By focusing solely on the first 5 mm of the contour, this
issue is mitigated, and the data are less noisy at these data points. In
figure SS5 the bond-bond correlation curves are reported.

####  I.3.2 Effect of the geometry on $\tau_{e}$ and $R_{e}$

In order to use the prediction for the long-time diffusion of tangentially
driven chain [17], we determine the average end-to-end distance $R_{e}$ and
the reorientational decorrelation time $\tau_{e}$, both obtained from the
detection and tracking of the worm’s contour as mentioned above. Sup. Figs.S
S6(a)-(c) show the orientational correlation in time, the average
reorientational decorrelation time and the average end-to-end distance of the
worms across all geometries, respectively. It is notable that we observe no
significant dependence of the pillar surface fraction $\phi$ on $\tau_{e}$,
while only a weak dependence is observed for $R_{e}$. This discrepancy may
seem surprising, especially given the strong dependence of the persistence
length of the worm on $\phi$ (as discussed in Section C1 above and depicted in
Fig. 2(a) of the main text). However, this inconsistency can be attributed to
the difficulty of accurately measuring $R_{e}$ and $\tau_{e}$ when parts of
the worm are obscured by the pillars. Unlike the tracking of the center of
mass and the determination of the persistence length, which remain well
defined despite partial obstruction, identifying the endpoints of the worm’s
skeleton for $R_{e}$ and $\tau_{e}$ measurements can be difficult when
obscured by the pillars.

Figure S6: Effect of the lattice geometry on $\tau_{e}$ and $R_{e}$. (a) The
orientational correlation in time, (b) the average reorientational
decorrelation time and (c) the average end-to-end distance of the worms across
all geometries.

###  I.4 Locomotion

####  I.4.1 Run lengths and trapping times

In the main text, we excluded the points for the ordered lattice at $50\%$
because the data for this experiments deviates slightly from the trend. The
reasons why are easily explained, but distract from the main point of the
figure. The ordered lattice at $50\%$ has a non-monotonous pore size
distribution because this is a slightly noisy hexagonal lattice with a few
removed pillars (see figure SS1&SS2). The worms still show hopping-trapping
behavior where they hop through very short tubes between pores, but the worms
are sometimes able to find longer tubes while simultaneously having longer
trapping times. In figure S7 the values of the run length and trapping times
of this data are reported

Figure S7: Tube length and trapping time distributions (a) Worms are defined
as caged when their instantaneous speed drops below a cutoff of $0.5~{}mm/s$.
Between caging events, worms crawl effective tubes of length $L_{tube}$. (b)
Example of a worm trajectory ($\phi=40\%$) as it reptates within an effective
tube made by the disordered positioning of pillars. (c) and (d) Distribution
of tube lengths for disordered (c) and ordered (d) media. Longer tube lengths
are observed as $\phi$ increases in the disordered case, while the maximum
tube length decreases for higher $\phi$ in the ordered medium. The opposite
trend is observed for the distribution of trapping times in the disordered
(e), and ordered media (f). The half-open symbols correspond to the ordered
lattice at $\phi=50\%$.

####  I.4.2 Effect of temperature

When placed in warmer water, our living worms become more active, resulting in
faster motion and increased shape’s fluctuation rates. In free space, their
long-time diffusion increases with temperature [21]. However, in dense
disordered media, both the long-time diffusion time and the reorientational
relaxation time decrease at higher temperatures. This occurs because the worms
become trapped in cavities more frequently, preventing them from being
stretched out long enough to initiate reptation. This trend is also evident in
the effective persistence length of the worms. See figure SS8 for the MSD
curves in laboratory units, Figure SS9 (a) for the persistence length and
figure SS9(b) for the calculation of the reorientational relaxation time.

Figure S8: MSD for the same set of worms at different temperatures in lab
units. a) Worms in free space. The hotter the worms, the more active they are
and the higher their MSD-curves b) Worms in $\phi=40\%$ disordered porous
media. The cold worms show strong reptation (even stronger than worms at room
temperature), while the worms in a $30\celsius$ bath never reptate and show
purely diffusive hopping-trapping behavior. Figure S9: Dependence of $l_{p}$
and $\tau_{e}$ on the temperature a) bond-bond correlation for worms at
$5\celsius$ (blue) and $30\celsius$ (red) as a function of time in free space
(dotted lines) and the disordered media at $\phi$=40% (solid lines). The
intersection with the black dashed line at $1/e$ defines the persistence
length. b) Orientational correlation for worms at $5\celsius$ (blue) and
$30\celsius$ (red) as a function of time in free space (dotted lines) and the
disordered media at $\phi$=40% (solid lines). The intersection with the black
dashed line at $1/e$ defines the reorientational relaxation time.

####  I.4.3 Bimodal distributions in the disordered geometries

Here we would like to zoom in a bit more on the trajectories and subsequent
MSD curves of the worms in the disordered geometries. Interestingly, distinct
trajectories emerge in our experiments, segregating into two populations
across all obstacle densities. One population showcases elongated, ballistic
stretches as the worms reptate from one tunnel to another. In contrast, the
second population exhibits hopping behavior, crawling from cavity to cavity
with purely diffusive dynamics. As the obstacle density increases, the
likelihood that a worm belongs to the ballistic population also increases.
Consequently, the long-time diffusion constant increases when averaging over
all trajectories from both populations, as illustrated in Figure SS10. In the
top panel of the figure, for each maze configuration, the trajectories are
segregated into two populations using a cutoff at the intermediate slope of
each trajectory. It is evident that the worms exhibit ballistic and diffusive
motion within each maze. However, as depicted in the PDFs in the bottom
panels, there is a notable shift in behavior, with worms transitioning from a
preference for diffusive motion to a preference for ballistic movement as
obstacle density increases. This shift is attributed to the worm’s capacity to
reptate through effective tubes mapped out by the position of the pillars. It
is worth noting here that the same individual worms were tested across all
mazes, indicating behavioral changes in individual worms.

Figure S10: average MSD curves and slope distribution for the disordered media
For the disordered maze, some worms travel in large ballistic stretches, while
some worms move along a diffusive trajectory. The probability of a worm moving
ballistically increases if the worm is in a more crowded environment (e.g.,
higher $\phi$).

##  II SIMULATIONS

###  II.1 Tangentially-driven polymer model

Figure S11: Schematic of the active tangentially-driven polymer in the ordered
and disordered arrangement of obstacles.

We implement the tangentially-driven polymer model [27] into both a 2D ordered
and disordered arrangement of circular obstacles, as illustrated in Sup. Fig.S
S11. In our experimental setup, we observe the 2D projection of 3D active
filaments around cylindrical pillars, indicating that the polymer can
intersect with itself. To take into account this behavior, we neglect excluded
volume interactions between monomers, and instead, we consider a phantom
active polymer model comprising $N$ monomers. The motion of each monomer
follows overdamped Langevin dynamics, described by:

$\gamma\dot{\vec{r}}_{i}=-\sum_{j}\nabla_{\vec{r}_{i}}U+\vec{f}^{a}_{i}+\vec{f}^{r}_{i},$
(1)

where $\vec{r}_{i}$ is the position of the $i$th monomer, the dot denotes the
derivative with respect to time and $\gamma$ is the friction coefficient
between the bead and its surrounding medium.

The potential energy $U$ of each monomer includes three different
contributions. The first one is the harmonic spring potential
$U_{\text{harmonic}}(r)=(k_{s}/2)(r-\ell)^{2}$, with equilibrium length $\ell$
and spring stiffness $k_{s}$ between adjacent monomers. The second part is the
bending potential between each two neighboring bonds
$U_{\text{bend}}(\theta_{i})=\kappa(1-cos\theta_{i})$, where $\theta_{i}$
denotes the angle between two consequent bonds intersecting at bead $i$
defined as
$\theta_{i}=\cos^{-1}(\widehat{{t}}_{i,i+1}\cdot\widehat{{t}}_{i-1,i})$ with
$\widehat{{t}}_{i,i+1}=\vec{r}_{i,i+1}/|\vec{r}_{i,i+1}|$ and
$\vec{r}_{i,i+1}=\vec{r}_{i+1}-\vec{r}_{i}$. Here, $\kappa$ is the bending
stiffness and determines the intrinsic degree of flexibility of a polymer.
Finally, the third contribution accounts for the excluded volume interactions
between each bead and its surrounding obstacles. They are modeled by the
short-ranged Weeks-Chandler-Andersen (WCA) potential [37]:

$U_{\text{excl}}(r)=4\epsilon\left[(\frac{\sigma/2+r_{o}}{r})^{12}-(\frac{\sigma/2+r_{o}}{r})^{6}+\frac{1}{4}\right]$
(2)

for $r<r_{c}=2^{1/6}(\sigma/2+r_{o})$, where $\epsilon$ is the strength of the
potential and has unit of energy, $\sigma$ is the diameter of the beads and
$r_{o}$ is the radius of obstacles. The WCA potential is zero for interaction
distances larger than the cutoff length $r_{c}$.

The active force on each bead, except for the end monomers, is given by:
$\vec{f}^{a}_{i}=\frac{f^{a}}{2\ell}(\vec{r}_{i-1,i}+\vec{r}_{i,i+1})$. The
active force on the tail monomer is given by
$\vec{f}^{a}_{1}=\frac{f^{a}}{2\ell}\vec{r}_{1,2}$ and for the head monomer by
$\vec{f}^{a}_{N}=\frac{f^{a}}{2\ell}\vec{r}_{N-1,N}$. The random force is
chosen as a white noise of zero mean and has the correlation
$\langle\vec{f}^{r}_{i}(t)\cdot\vec{f}^{r}_{j}(t^{\prime})\rangle=4D_{0}\gamma^{2}\delta_{ij}\delta(t-t^{\prime})$.
It should be noted that the persistence length of a 2D passive ideal polymer
in free space can be determined in terms of its bending stiffness and the
strength of random force correlation as $l_{p}^{0}=2\kappa\sigma/D_{0}\gamma$.
[38].

We use the coordinates of the pillars in experiment to position the obstacles
in a 2D simulation box with periodic boundary condition. We choose
$l_{u}=\sigma$, $E_{u}=\epsilon$ and $\tau_{u}=\gamma\sigma^{2}/\epsilon$ with
$\gamma=1$ as the units of length, energy, and time. Subsequently, we fix
$\ell=1\sigma$, $N=100$, $r_{o}=8.33\sigma$ and the diffusion coefficient
$D_{0}=1\epsilon/\gamma$. The ratio between the obstacle radius, chain length
and monomer diameter is set with respect to the average length and thickness
of the worms. We choose an active force of $f^{a}=0.1\epsilon/\sigma$ and the
spring constants are chosen very stiff $k_{s}=5000\epsilon/\sigma\gg
f^{a}/\ell$, to ensure that the mean bond length and the polymer contour
length remain almost constant during simulations. It has been well established
that the relaxation time of flexible tangentially driven chains in free space
scales as $\tau_{e}\sim 1/f^{a}$, while their enhanced diffusion coefficient
scales as $D_{l}\sim f^{a}$ [17, 29]. Hence, when using the relaxation time as
the unit of time, the $D_{l}$ of chains with different activities become
identical. Our choice of active force ($f^{a}=0.1$), ensures that the activity
dominates the motion of the whole polymer chain, but it is weak enough to let
the thermal fluctuation affect the chain on monomer level. At the chain level,
the thermal relaxation time suggested by the Rouse model is
$\tau^{\text{Rouse}}\sim N^{2}=10^{4}$ [30], which is $10$ times slower than
the active relaxation time $\tau^{\text{Active}}\sim N/f^{a}=10^{3}$. However,
for a segment of two monomers, the thermal time is $1$, whereas the active
time is $1/f^{a}=10$. It is worth mentioning that any active force of the same
order of magnitude would qualitatively give the same results as those given by
$f^{a}=0.1$.

###  II.2 MSD curves and input bending stiffness

To gain insights from the tangentially driven polymer model, we kept certain
input parameters fixed. In our simulations, spatial dimensions were determined
relative to the setup size and the average length and thickness of the worms,
as detailed earlier. Activity was intentionally set to low values to ensure
that the contour fluctuations of the active polymer mirrored the fluctuations
observed in the worms. Subsequently, the bending stiffness was derived from
the persistence length of the worms (as described above). However, due to
activity and interactions with obstacles (e.g. confinement), the effective
persistence length, and thus the bending stiffness of the tangentially driven
polymer, deviated from the input value. The bending stiffness was adjusted to
ensure that the effective persistence length matched between experiments and
simulations. The values are reported below:

Setup | $\kappa_{input}$ | $\kappa_{effective}$ | $\kappa_{experiments}$
---|---|---|---
Free space, $\phi=0$ | 9 | 9.0 | 9,17
Disordered, $\phi=10$ | 10 | 9.5 | 9.36
Disordered, $\phi=20$ | 10 | 9.5 | 9.31
Disordered, $\phi=30$ | 11 | 10.5 | 10.95
Disordered, $\phi=40$ | 13 | 12.0 | 11.95
Ordered, $\phi=40$ | 7 | 7.0 | 7.17
Ordered, $\phi=50$ | 5.5 | 6.0 | 6.20

In figure S12 the results from the simulations are shown, all curves are
rescaled by their respective rotational decorrelation time.

Figure S12: MSD results from the simulations. (a) All simulations are done in
the same geometries as the experiments, with the relevant bending stiffness
and rescaled by their respective rotational decorrelation time. See table II.2
for all bending stiffness used. (b) Zoom to the diffusive part of the MSD.

###  II.3 Characterization of trapping events for simulation

We measure trapping and reptating events based on the following method for the
simulated active polymers. Since for tangentially driven polymers the self
propulsion velocity of the center of the mass is proportional to $R_{e}$, we
use the distribution of the end-to-end distance to find the threshold on
$R_{e}$ bellow which the chain is trapped state. In Fig.S13 we have the
$P(R_{e})$ for all of the simulations. The most confined chain is the one in
the ordered medium with $\phi=0.5$. This particular distribution has three
peaks showing the interaction of the polymer with the obstacles. We use the
position of the first peak at $R_{e}\approx 25$ to set the threshold. We use
the same value for all of the simulations for consistency. Therefore, a chain
is labeled trapped when $R_{e}<25$, otherwise reptating.

Figure S13: The probability distribution function of end-to-end distance of
simulated polymers in different media. The black dashed line shows the value
of the first peak ($R_{e}=25$) of the active polymer moving in ordered medium
with $\phi=0.5$.