# Structural dynamics and optimal transport of an active polymer

Hamidreza Khalilian<EMAIL_ADDRESS>School of Nano sciences, Institute for
Research in Fundamental Sciences (IPM), 19395-5531, Tehran, Iran.  Fernando
Peruani<EMAIL_ADDRESS>Laboratoire de Physique Théorique et
Modélisation, UMR 8089, CY Cergy Paris Université, 95302 Cergy-Pontoise,
France.  Jalal Sarabadani<EMAIL_ADDRESS>School of Nano sciences, Institute for
Research in Fundamental Sciences (IPM), 19395-5531, Tehran, Iran.

###### Abstract

We study the spontaneous configuration transitions of an active semi-flexible
polymer between spiral and non-spiral states, and show that the configuration
dynamics is fully described by a subcritical pitchfork bifurcation. Exploiting
the fact that active polymer barely moves in spiral states and exhibits net
displacements in non-spiral states, we prove that the motion of the active
polymer is consistent with a run-and-tumble-like dynamics. Moreover, we find
that there exists an optimal self-propelling force, at which the probabilities
of finding the polymer in the spiral and non-spiral state become equal, that
maximizes the diffusion coefficient.

Active polymers have recently become an interesting field of research in soft
matter and non-equilibrium statistical physics. They play an important role in
biological systems such as duplication of DNA by DNA-polymerase albert ,
synthesizes of proteins by ribosomes albert ; zia , chromatin in eukaryotic
cells chromatin1 ; chromatin2 ; chromatin3 ; chromatin4 ; chromatin5 ;
chromatin6 , movement of actin filaments by motor proteins in cellular
cytoskeletons cytos1 ; cytos2 ; turning6 ; turning7  and the oscillatory
beating to drive of flagella flagella1 ; flagella2 . In all of the above
examples, as mechanical tangential forces are generated on the backbone of the
objects, they can be considered as active polymers. Motivated by these
applications, many studies have focused, both experimentally exp1 ; exp2 ;
exp3 ; exp4 ; exp5  and theoretically theo1 ; theo2 ; theo3 ; theo4 ; theo5 ;
theo6 ; theo7 ; theo8 ; theo9 ; PRL ; theo10 ; baskaran ; theo11 ; turning ;
theo12 , on active polymers. Theoretical studies have shown that the diffusion
coefficient of the flexible/semi-flexible active polymer is a monotonically
increasing function – linear or quadratic – of Péclet number theo1 ; theo5 ;
theo6 ; theo7 ; theo9 ; PRL ; baskaran ; theo11 ; theo12 .

The structure and configuration of an active polymer have been studied in
theo4 ; theo5 ; theo6 ; theo9 ; PRL ; theo10 ; baskaran ; turning ; theo11 .
The interplay between thermal fluctuations and self-propelling stresses (i.e.
activity) theo9 ; baskaran ; turning  – as occurs when a polymer is embedded
in a gliding assay with motor proteins as the source of activity theo10 ;
turning  – leads to transient polymer configurations, classified into two main
groups: spiral and non-spiral ones. For the latter case turning  the radius
of gyration (RG) is a convex function with respect to the
attachment/detachment rates and Péclet number of the motor proteins, while for
a system composed of an active polymer wherein the force is applied to all
beads of the polymer theo9  RG is a monotonic decreasing function of the
polymer activity[see the Supplemental Material (SM) for more details].

Figure 1: (a) Typical snapshot of an active semi-flexible polymer with contour
length of $N=200$ and persistence length of $\ell_{\textrm{p}}=5$. The polymer
is divided to $N/\ell_{\textrm{p}}$ segments. The red and green beads
represent the tail and head of each segment, respectively. In each segment the
self-propelling force $F_{\textrm{sp}}$ (orange arrow), acts on the head
monomer directing from tail to head (orange dashed line). (b) Definition of
the spiral and non-spiral states: probability distribution function of the
turning number $P(\psi)$ for fixed values of $F_{\textrm{sp}}=2$, $N=200$ and
$\ell_{\textrm{p}}=5$. The vertical black dashed lines at minima of the
$P(\psi)$ (located at the left $\psi_{\textrm{min}}^{\textrm{l}}$ and the
right $\psi_{\textrm{min}}^{\textrm{r}}$ minima) separate the two spiral and
non-spiral states. The polymer with
$\psi_{\textrm{min}}^{\textrm{l}}<\psi<\psi_{\textrm{min}}^{\textrm{r}}$ is in
the non-spiral state, and it is in the spiral state for
$\psi<\psi_{\textrm{min}}^{\textrm{l}}$ or
$\psi>\psi_{\textrm{min}}^{\textrm{r}}$. The right and left snapshots
correspond to the spiral states with counter-clockwise and clockwise turnings,
respectively, and the middle one exhibits a non-spiral state. See SM for
movies.

In the present paper, we show that transitions between configurational states
of the polymer is consistent with a subcritical pitchfork bifurcation.
Furthermore, we prove that the diffusion coefficient for the polymer center of
mass (CM), $D_{\textrm{CM}}$ is, counterintuitively, a non-monotonic function
of the self-propelling (SP) force, $F_{\textrm{sp}}$, acting on different
segments of the polymer. Moreover, we find that there exist an optimal self-
propelling force $F_{\textrm{sp}}$ that maximizes $D_{\textrm{CM}}$.

Simulation model –  We consider a polymer with $N=200$ monomers in a two-
dimensional, squared simulation box of area $L\\!\times\\!L$ – where
$L=500\sigma$ and $\sigma$ the length unit – with periodic boundary conditions
[Fig. 1(a)]. The $N$ monomers are disks of radius $\sigma$ and mass M. The
consecutive monomers are connected by the finitely extensible nonlinear
elastic (FENE) potential
$U_{\textrm{FENE}}(r)=-\frac{1}{2}kR^{2}_{0}\ln[1-(r/R_{0})^{2}]$, where $r$
is the center-to-center distance between two connected monomers, $k$ the
spring constant, and $R_{0}$ the natural length of the spring. Monomers
interact via the repulsive Weeks-Chandler-Anderson (WCA) wca  potential
$U_{\textrm{WCA}}(r)=U_{\textrm{LJ}}(r)-U_{\textrm{LJ}}(r_{\textrm{c}})$ if
$r\leq r_{\textrm{c}}$ and zero otherwise, where $r_{c}=2^{1/6}\sigma$ and $r$
are the cut-off radius and the distance between two given monomers,
respectively.
$U_{\textrm{LJ}}(r)=4\epsilon\big{[}\big{(}\sigma/r\big{)}^{12}-\big{(}\sigma/r\big{)}^{6}\big{]}$
is the Lennard-Jones (LJ) potential, with $\epsilon$ as the depth of the
potential well. The rigidity of the semi-flexible polymer is modeled by the
cosine potential
$U_{\textrm{bend}}=\kappa_{\textrm{b}}[1+\cos\left(\theta_{i+1}-\theta_{i}\right)]$,
where $\kappa_{\textrm{b}}$ is a bending constant, $\theta_{i}$ the angle
between the $i^{\textrm{th}}$ bond vector and horizontal axis, and
$\theta_{i+1}-\theta_{i}$ represents the angle between two consecutive bond
vectors, whose positive and negative values correspond to the counter-
clockwise (CCW) and clockwise (CW) local configurations of the consecutive
$i^{\textrm{th}}$ and $(i+1)^{\textrm{th}}$ bonds, respectively. For non-zero
value of the $\kappa_{\textrm{b}}$, the persistence length of the polymer in
two dimensions is obtained as
$\ell_{\textrm{p}}=2\kappa_{\textrm{b}}/(k_{\textrm{B}}T)$. The
$k_{\textrm{B}}T$ is the thermal energy and $k_{\textrm{B}}$ is the Boltzmann
constant. To model the active polymer, the polymer is divided into
$N/\ell_{\textrm{{p}}}$ segments. Each segment – see Fig. 1(a) – contains a
tail (in red color) and a head (in green color) monomers as the first and last
monomers, respectively, and the rest of monomers are in blue color.
Importantly, a self-propelling force $F_{\textrm{sp}}$ acts on the head
monomer of each segment and in the direction from tail to head, depicted by
the orange arrow and dashed line, respectively in Fig. 1(a) Amir .

Employing Langevin dynamics (LD), we express the temporal evolution of the
$i$th monomer by the equation of motion:
$M\ddot{\vec{r}}_{i}=-\gamma\dot{\vec{r}}_{i}+\vec{F}_{\textrm{sp}}\delta_{\textrm{ih}}-\vec{\nabla}_{r}U_{\textrm{tot}}+\sqrt{2\gamma
k_{\textrm{B}}T}\vec{\eta}_{i}(t)$, where
$U_{\textrm{tot}}=U_{\textrm{FENE}}+U_{\textrm{bend}}+U_{\textrm{LJ}}$ is the
total potential energy and $\gamma$ is the solvent friction coefficient that
the $i$th monomer experiences. In the Kronecker delta $\delta_{\textrm{ih}}$
the h stands as an index of the head monomer in each segment. The white noise
term follows $\langle\eta_{i}^{m}(t)\rangle=0$ and
$\langle\eta_{i}^{m}(t)\eta_{j}^{l}(t^{\prime})\rangle=\delta_{ij}\delta_{ml}\delta(t-t^{\prime})$,
where $m,l\equiv x,y$. The M, $\sigma$ and $\epsilon$ are used as the
simulation unit scales for mass, length and energy, respectively. The
temperature is kept at $k_{\textrm{B}}T=1.2\epsilon$ and the friction
coefficient is set to $\gamma=0.7\tau_{0}^{-1}$, in which
$\tau_{0}=\sqrt{\textrm{M}\sigma^{2}/\epsilon}$ is the LJ time unit. In our
simulations, we consider $dt=0.001\tau_{0}$ and $t_{\textrm{eq}}=5\times
10^{4}\tau_{0}$ as the time step for the integration of the equations of
motion and the equilibration time interval, respectively. We also set $k=30$,
$R_{0}=1.5$ and $\kappa_{\textrm{b}}=3$ to have a semi-flexible polymer with
persistence length of $\ell_{\textrm{p}}=5$. The polymer is initially
equilibrated during the time interval $t_{\textrm{eq}}$ with
$F_{\textrm{sp}}=0$. Then the SP force is switched on and the main simulations
are done for $10^{6}\tau_{0}$. The results are obtained by averaging over
$10^{6}$ data points from 10 different trajectories. All the LD simulations
are performed using LAMMPS package lammps .

Figure 2: (a) Non-monotonic diffusion coefficient of the polymer center of
mass $D_{\textrm{CM}}$ as a function of the SP force $F_{\textrm{sp}}$, from
LD simulations (LD Sim; black circles) and the theory obtained from Eq. (4)
(Theory; orange squares). (b) The probability of finding the polymer in the
non-spiral state $p_{\textrm{ns}}$ (red squares) and in the spiral state
$p_{\textrm{s}}$ (blue circles) as a function of the SP force
$F_{\textrm{sp}}$. The intersection of the red and blue curves (denoted by the
horizontal black dashed line at $p_{\textrm{ns}}=p_{\textrm{s}}=0.5$ and the
vertical black dashed-dotted line at $F_{\textrm{sp}}=1.9$) corresponds to the
peak of the black curve $D_{\textrm{CM}}$ in panel (a).

Spiral states & transport –  As the snapshots in Fig. 1(b) show, the self-
propelled polymer possesses different configurations that correspond to spiral
and non-spiral states, which are quantified by considering the turning number
$\psi$ of the polymer. The turning number of the entire polymer with $N$ bonds
is defined as turn_handbook

$\psi=\frac{1}{2\pi}\sum^{N-1}_{i=1}(\theta_{i+1}-\theta_{i})\,.$ (1)

Note that $\psi>0$ [right snapshot in Fig. 1(b)] and $\psi<0$ [left snapshot
in Fig. 1(b)] imply that the global configuration of the polymer is CCW and
CW, respectively. A vanishing $\psi$ value indicates either a rod-like
configuration or a random coil polymer configuration [middle snapshot in Fig.
1(b)].

In simulations, we find that the probability distribution function (PDF) of
the turning number $P(\psi)$ has three peaks: two are located in the left and
right hand sides and one at $\psi=0$; see Fig. 1(b) where two vertical black
dashed lines, at the two minima of the $P(\psi)$, indicate the region of
$\psi$ that corresponds to non-spiral states. In the spiral state, the active
forces acting on the monomers cancel out and thus the polymer barely moves. On
the other hand, when the active polymer escapes from a spiral configuration
and stays in the non-spiral state, it is less coiled and active forces along
the polymer contribute to displace the center of mass. Fig. 2(a) shows the
$D_{\textrm{CM}}$ as a function of $F_{\textrm{sp}}$ for a fixed value of
$\ell_{\textrm{p}}=5$ (black circles), where it is evident that
$D_{\textrm{CM}}$ is non-monotonic with respect to the $F_{\textrm{sp}}$.
Moreover, it exists an optimal value of $F_{\textrm{sp}}$ that maximizes
$D_{\textrm{CM}}$. The maximum of $D_{\textrm{CM}}$ takes place when the
probabilities of finding the polymer in the spiral and non-spiral states are
equal to each other; see Fig. 2(b).

Figure 3: (a) The PDF of the $\psi$, $P(\psi)$ obtained from integration of
Eq. (2) with $\chi=-0.184$ corresponding to Sp force $F_{\textrm{sp}}=1.9$
and for fixed value of the noise $\Gamma=0.00375$ (orange circles) and from
the LD simulations (purple squared). (b) Bifurcation diagram: theory vs.
simulation. The blue solid and dashed lines that are obtained from Eq. (3)
show the stable and unstable fixed points, respectively. The filled and empty
green circles represent the stable and unstable fixed points, respectively,
and are coming from the LD simulations. Inset presents the value of
$F_{\textrm{sp}}$ as a function of $\chi$. (c) The PDF of $x$, $P(x)$ (top)
and the corresponding potential $\Phi(x)$ (bottom). In both panels the value
of $\chi$ has been set to $\chi=-0.2$. (d) The probability of finding the
polymer in the non-spiral state $p_{\textrm{ns}}$ (red squares from the LD
simulation and red solid line from the theory) and in the spiral state
$p_{\textrm{s}}$ (blue circles from the LD simulation and blue solid line from
the theory) as a function of $\chi$. The intersection of the red and blue
curves (denoted by the horizontal black dashed line at
$p_{\textrm{ns}}=p_{\textrm{s}}=0.5$ and the vertical black dashed-dotted line
at $\chi=-0.184$ corresponds to $F_{\textrm{sp}}=1.9$) corresponds to the peak
of the black curve $D_{\textrm{CM}}$ in Fig. 2(a).

These probabilities are defined as
$p_{\textrm{s}}=\frac{\langle\tau_{\textrm{s}}\rangle}{\langle\tau_{\textrm{s}}\rangle+\langle\tau_{\textrm{ns}}\rangle}$
and
$p_{\textrm{ns}}=\frac{\langle\tau_{\textrm{ns}}\rangle}{\langle\tau_{\textrm{s}}\rangle+\langle\tau_{\textrm{ns}}\rangle}$,
where $\langle\tau_{\textrm{s}}\rangle$ and $\langle\tau_{\textrm{ns}}\rangle$
refer to the average time the polymer stays in the spiral and non-spiral
states, respectively.

Theory –  The PDF of the turning number $\psi$ can be expressed as
$P(\psi)\sim\textrm{exp}[-\Phi(\psi)]$ with
$\Phi(\psi)=a_{2}\psi^{2}-a_{4}\psi^{4}+a_{6}\psi^{6}$ a potential, where
$a_{2},a_{4}$ and $a_{6}$ are constants that depend on the value of
$F_{\textrm{sp}}$ [see Fig. 1(b) and in SM, Fig. 1(b)]; cf. turning .
Defining an effective “force” as $f(\psi)=-d\Phi(\psi)/d\psi$, the dynamics of
$\psi$ can be described in dimensionless units as

$\frac{dx}{d\tau}=f(x)+\xi({\tau})=\chi x+x^{3}-x^{5}+\xi({\tau}),$ (2)

where $x=\psi/U$ and $\tau=t/\mathcal{T}$ with $U^{2}=2a_{4}/(3a_{6})$,
$\mathcal{T}=3a_{6}/(8a^{2}_{4})$ and $\chi=-3a_{2}a_{6}/(4a^{2}_{4})$. The
term $\xi$ refers to a white noise that satisfies $\langle\xi(\tau)\rangle=0$
and
$\langle\xi(\tau)\xi(\tau^{{}^{\prime}})\rangle=2\Gamma\delta(\tau-\tau^{{}^{\prime}})$.
The noise intensity $\Gamma$ is obtained by mapping the PDF of the turning
number in LD simulations with the one resulting from the integration of Eq.
(2) [see Fig. 3(a)]. The deterministic form of Eq. (2) – that corresponds
to $\Gamma=0$ and thus $\dot{x}=f(x;\chi)$ – undergoes a subcritical pitchfork
bifurcation strogatz . The bifurcation diagram in the $x-\chi$ plane is shown
in Fig. 3(b). The fixed points of Eq. (2) – $f(x^{*};\chi)=0$ – are:

$x^{*}=0,\,\pm\bigg{(}\frac{1\pm\sqrt{1+4\chi}}{2}\bigg{)}^{1/2}\,.$ (3)

For $\chi>0$, $x^{*}=0$ is an (linearly) unstable fixed point since
$f^{{}^{\prime}}(x^{*})=\frac{\partial f}{\partial x}(x^{*})\geq 0$. For
$\chi\leq 0$, $x^{*}=0$ becomes stable ($f^{{}^{\prime}}(0)<0$) and two
unstable fixed points – such that $f^{{}^{\prime}}(x^{*})\geq 0$ – emerge. At
$\chi_{\textrm{s}}=-1/4$ there are two additional saddle-node bifurcations
[denoted by a black star in the main panel of Fig. 3(b)] which are obtained
by considering the condition $1+4\chi_{\textrm{s}}\geq 0$. Note that the fixed
points are the extremum values of the $P(\psi)$ [see Fig. 1(b) in SM]. The
effect of $x^{5}$ term in Eq. (2) in the bifurcation diagram is to turn
around the unstable branches at $\chi=\chi_{\textrm{s}}$ and become stable for
$\chi>\chi_{\textrm{s}}$ [solid blue lines in the top and bottom in Fig. 3(b)
main panel]. These stable states provide the possibility of jumps and
hysteresis as $\chi$ is varied, that is the characteristic of the first-order
phase transitions. The filled and empty green circles in Fig. 3(b) correspond
to the LD simulations and show the stable and unstable states, respectively.
The simulations data and theoretical curves show a perfect agreement. The
inset in Fig. 3(b) shows how the values of $F_{\textrm{sp}}$ and $\chi$
relate to each other.

To compute the diffusion coefficient $D_{\textrm{CM}}$, and to put in evidence
the run-&-tumble character of the process, we consider that during the spiral
states, the sum of active forces acting on the monomers cancels out. Thus, in
this state, the speed (of the CM) of polymer is $v_{\textrm{s}}\simeq 0$. This
implies that we assume that the distance travelled by the polymer in the
spiral sates is negligible [see the movie in SM]. On the other hand, in the
non-spiral states, the polymer is (in comparison to spiral states) extended,
and thus, the sum of active forces is non-zero. In consequence, in the non-
spiral state, the polymer moves at speed $v_{\textrm{ns}}>0$, and we assume
that it travels an average distance
$\ell=v_{\textrm{ns}}\langle\tau_{\textrm{ns}}\rangle$. To determine
$v_{\textrm{ns}}$, let us consider that in the non-spiral state, the polymer
is an extended rod – that experiences an active force
$NF_{\textrm{sp}}/\ell_{\textrm{p}}$ and a friction drag $N\,\gamma$ – and
thus, moves at speed
$v_{\textrm{ns}}=F_{\textrm{sp}}/\gamma~{}\\!\ell_{\textrm{p}}$. All together,
this implies that as the polymer goes through a cycle spiral-non-spiral, i.e.
involving a transition from the spiral to the non-spiral state, moves an
average distance $\ell$. Since the average duration of these cycles is
$\Omega=\langle\tau_{\textrm{s}}\rangle+\langle\tau_{\textrm{ns}}\rangle$, in
a time $t$, the polymer performs an average of $n=t/\Omega$ of those cycles.
It is worth noting that $\langle\tau_{\textrm{ns}}\rangle$ and
$\langle\tau_{\textrm{s}}\rangle$ can be estimated from the transition rates
to escape from the non-spiral and spiral states considering the associated
Kramer’s escape problem that assumes these rates are proportional to the
exponential of the potential barrier:
$\langle\tau_{\textrm{ns}}\rangle^{-1}\propto\exp(\Delta\Phi_{1}/\Gamma)$ and
$\langle\tau_{\textrm{s}}\rangle^{-1}\propto\exp(\Delta\Phi_{2}/\Gamma)$,
respectively, where $\Delta\Phi_{1}=\frac{1}{12}(1+4\chi)^{3/2}$ and
$\Delta\Phi_{2}=\frac{1}{48}(-1+\sqrt{1+4\chi})(1+8\chi-\sqrt{1+4\chi})$ [see
Fig. 3(c) bottom panel]. Recall that then $p_{\textrm{s}}$ and
$p_{\textrm{ns}}$ can also be expressed in term of these rates, and that
$\chi=\chi(F_{\textrm{sp}})$ [see Fig. 3(d)]. Finally, let us recall that for
a random walk (RW) after $n_{\textrm{RW}}$ steps of length
$\ell_{\textrm{RW}}$, the mean-square displacement goes as
$\langle\mathbf{r}^{2}\rangle=n_{\textrm{RW}}\,\ell_{\textrm{RW}}^{2}$. Thus,
if we interpret that in time $t$, the $n$ cycles performed by the active
polymer correspond to $n_{\textrm{RW}}$ steps of a RW, that moves in each step
an average distance $\ell_{\textrm{RW}}=\ell$, we can approximate the
diffusion coefficient – using the above-given expressions for $n$ and $\ell$ –
as:

$D_{\textrm{CM}}=\lim_{t\to\infty}\frac{\langle\mathbf{r}^{2}\rangle}{4t}=\frac{v_{\textrm{ns}}^{2}}{4}\frac{\langle\tau_{\textrm{ns}}\rangle^{2}}{\langle\tau_{\textrm{s}}\rangle+\langle\tau_{\textrm{ns}}\rangle}=\frac{\overline{v}^{2}\Omega}{4}\,,$
(4)

where $\overline{v}=p_{\textrm{ns}}v_{\textrm{ns}}$. The comparison between
the diffusion coefficients $D_{\textrm{CM}}$ predicted by Eq. (4) (denoted by
”Theory”; orange squares) and the one obtained from the LD simulations (”LD
Sim”; black circles) displayed in Fig. 2(a), shows that there exists a good
quantitative agreement.

Importantly, Eqs. (2) and (4) allow us to prove that there is an optimal
self-propelling force that maximizes the diffusion coefficient. The existence
of an optimal self-propelling force is easy to understand. When
$F_{\textrm{sp}}\to 0$, the spiral states is not observed. The speed in the
non-spiral state is $v_{0}\propto F_{\textrm{sp}}$, then, $F_{\textrm{sp}}\to
0$ implies $D_{\textrm{CM}}\to 0$. On the contrary, for $F_{\textrm{sp}}\gg
1$, the polymer is locked in spirals, but since we assume that $v_{0}=0$ in
this state, we get again $D_{\textrm{CM}}\to 0$. In summary, increasing
$F_{\textrm{sp}}$ increases $v_{0}$, but at the same time enhances the
probability of finding the polymer in the spiral state, where the motion of
the center of mass of the polymer is strongly reduced. And thus, we find that
for intermediate values of $F_{\textrm{sp}}$, $D_{\textrm{CM}}$ is maximized.

Our studies puts in evidence the existence of a key coupling between the
structural dynamics of the polymer and its transport properties. Moreover, we
have shown that there exist an optimal active force $F_{\textrm{sp}}$ that
maximizes the diffusion coefficient $D_{\textrm{CM}}$. The reported results
are of relevance for a large number of experimental active polymer studies,
including actin filament motion in motility assays experiments and DNA
duplication, among many other examples.

## References

  * (1) B. Alberts, J. Alexander, L. Julian, R. Martin, R. Keith, and W. Peter, Molecular Biology of the Cell (Garland Science, New York, 2002).
  * (2) R. K. P. Zia, J. J. Dong, and B. Schmittmann, J. Stat. Phys. 144, 405 (2011).
  * (3) I. Bronstein, Y. Israel, E. Kepten, S. Mai, Y. Shav-Tal, E. Barkai, and Y. Garini, Phys. Rev. Lett. 103, 018102 (2009).
  * (4) I. Bronshtein, E. Kepten, I. Kanter, S. Berezin, M. Lind- ner, A. B. Redwood, S. Mai, S. Gonzalo, R. Foisner, Y. Shav-Tal, and Y. Garini, Nat . Commun. 6, 8044 (2015).
  * (5) G. G. Cabal, A. Genovesio, S. Rodriguez-Navarro, C. Zimmer, O. Gadal, A. Lesne, H. Buc, F. Feuerbach- Fournier, J.-C. Olivo-Marin, E. C. Hurt, and U. Nehrbass, Nature 441, 770 (2006).
  * (6) A. Zidovska, D. A. Weitz, and T. J. Mitchison, Proc. Natl. Acad. Sci. 110, 15555 (2013).
  * (7) N. Ganai, S. Sengupta, and G. I. Menon, Nucleic Acids Res. 42, 4145 (2014).
  * (8) N. Haddad, D. Jost, and C. Vaillant, Chromosome Research 25, 35 (2017).
  * (9) C. P. Brangwynne, G. H. Koenderink, F. C. MacKintosh, and D. A. Weitz, Phys. Rev. Lett. 100, 118104 (2008).
  * (10) D. Mizuno, C. Tardin, C. F. Schmidt, and F. C. MacKintosh, Science 315, 370 (2007).
  * (11) D. A. Fletcher and R. D. Mullins, Nature 463, 485(2010).
  * (12) F. Jülicher, S. W. Grill, and G. Salbreux, Reports Prog. Phys. 81, 076601 (2018).
  * (13) R. Chelakkot, A. Gopinath, L. Mahadevan, and M. F. Hagan, J. R. Soc., Interface 11, 20130884 (2014).
  * (14) J. Elgeti, R. G. Winkler, and G. Gompper, Rep. Prog. Phys. 78, 056601 (2015).
  * (15) R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone, J. Bibette, Nature 437, 862 (2005).
  * (16) V. Schaller, C. Weber, C. Semmrich, E. Frey, A. R. Bausch, Nature 467, 73 (2010).
  * (17) Y. Sumino, K. H. Nagai, Y. Shitaka, D. Tanaka, K. Yoshikawa, H. Chaté and K. Oiwa, Nature 483, 448 (2012).
  * (18) L. J. Hill, N. E. Richey, Y. Sung, P. T. Dirlam, J. J. Griebel, E. Lavoie-Higgins, I.-B. Shim, N. Pinna, M.-G. Willinger, W. Vogel, J. J. Benkoski, K. Char, J Pyun, ACS Nano 8, 3272 (2014).
  * (19) R. Suzuki and A. R. Bausch, Nat. Commun. 8, 41 (2017).
  * (20) T. B. Liverpool, Phys. Rev. E 67, 031909 (2003).
  * (21) H. Jiang and Z. Hou, Soft Matter 10, 1012 (2014).
  * (22) A. Ghosh and N. Gov, Biophys. J. 107, 1065 (2014).
  * (23) R. E. Isele-Holder, J. Elgeti and G. Gompper, Soft Matter, 11, 7181 (2015).
  * (24) D. Sarkar, S. Thakur, Phys. Rev. E 93, 032508 (2016).
  * (25) T. Eisenstecken, G. Gompper and R. G. Winkler, Polymers 8, 304 (2016).
  * (26) T. Eisensteckena, G. Gompper and R. G. Winklera, J. Chem. Phys. 146, 154903 (2017).
  * (27) D. Osmanović and Y. Rabin, Soft Matter 13, 963 (2017).
  * (28) S. K. Anand and S. P. Singh, Phys. Rev. E 98, 042501 (2018).
  * (29) V. Bianco, E. Locatelli, and P. Malgaretti, Phys. Rev. Lett. 121, 217802 (2018).
  * (30) N. Gupta, A. Chaudhuri and D. Chaudhuri, Phys. Rev. E 99, 042405 (2019).
  * (31) M. S. E. Peterson, M. F. Hagan and A. Baskaran1. J. Stat. Mech. 2020, 013216, (2020).
  * (32) R. G. Winkler and G. Gompper, J. Chem. Phys. 153, 040901 (2020).
  * (33) A. Shee, N. Gupta, A. Chaudhuri and D. Chaudhuri, Soft Matter, 17, 2120 (2021).
  * (34) C. A. Philipps, G. Gompper and R. G. Winkler, J. Chem. Phys. 157, 194904 (2022).
  * (35) J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys. 54, 5237 (1971).
  * (36) A. Rezaie-Dereshgi, H. Khalilian and J. Sarabadani, J. Phys.: Condens. Matter, 35, 355101 (2023).
  * (37) S. Plimpton, J. Comp. Phys. 117, 1-19 (1995), http://lammps.sandia.gov.
  * (38) S. G. Krantz, Handbook of Complex Variables, (Birkhäuser, Boston, MA, 1999).
  * (39) Steven H. Strogatz, Nonlinear Dynamics and Chaos, (Perseus Books Publishing, L.L.c., 1994).