# Translocation of an Active Polymer into a Circular Cavity

Amir Rezaie-Dereshgi  School of Nano Science, Institute for Research in
Fundamental Sciences (IPM), 19395-5531, Tehran, Iran.  Department of Physics,
Institute for Advanced Studies in Basic Sciences (IASBS), 45137-66731, Zanjan,
Iran.  Hamidreza Khalilian  School of Nano Science, Institute for Research in
Fundamental Sciences (IPM), 19395-5531, Tehran, Iran.  Jalal Sarabadani
<EMAIL_ADDRESS>School of Nano Science, Institute for Research in Fundamental
Sciences (IPM), 19395-5531, Tehran, Iran.

###### Abstract

Translocation dynamics of an active semi-flexible polymer through a nano-pore
into a rigid two dimensional circular cavity, and the polymer packing dynamics
have been studied by using Langevin dynamics (LD) simulations. The results
show that the force exponent $\beta$, for regime of small cavity radius, i.e.
$R\ll R_{\textrm{g}}$, where $R_{\textrm{g}}$ is the gyration radius of the
passive semi-flexible polymer in two dimensional free space, is $\beta=-1$,
while for large values of $R\gg R_{\textrm{g}}$ the asymptotic value of the
force exponent is $\beta\approx-0.92$. The force exponent is defined by the
scaling form of the average translocation time $\langle\tau\rangle\propto
F_{\textrm{sp}}^{\beta}$, where $F_{\textrm{sp}}$ is the self-propelling
force. Moreover, using the definition of the turning number for the polymer
inside the cavity, it has been found that at the end of translocation process
for small value of $R$ and in the strong force limit the polymer configuration
is more regular than the case in which the value of $R$ is large or the force
is weak.

##  I Introduction

Polymer translocation through a nano-pore, which makes a bridge between
physics and biology, has attracted a lot of attention since the seminal
experimental bezrukov1994counting ; kasianowicz1996characterization  and
theoretical works sung1996polymer  in about three decades ago. Translocation
of a DNA and RNA through a nano-pore on a nuclear membrane, virus genome
injection and transportation of a protein chain across a pore on a membrane of
a cell, are some important examples of polymer translocation in biology
muthukumar2016polymer . Moreover, polymer translocation through a nano-pore
has been used to develop some nano-technological tools to manipulate a single
molecule in experiments, and a long cylindrical nano-pore setup has been used
to visualize the genomic information by investigation the DNA spatial
conformations Tegenfeldt_book ; Tegenfeldt_CSR2010 ; Tegenfeldt_ABC2004 ;
Tegenfeldt_PNAS2005 ; Tegenfeldt_PNAS2004 . Such a diverse and extensive
subject, from essential role of polymer translocation in the cell function to
its experimental applications has attracted people to study this phenomenon.
An important factor that affects the translocation dynamics is the geometry of
the trans side, i.e. the available space for the polymer in the trans side,
which can be semi-space sung1996polymer ; bhattacharya2009scaling ;
luo2007heteropolymer ; luo2008dynamics ; rowghanian2011force ;
saito2011dynamical ; saito2012process ; sakaue2007nonequilibrium ;
sakaue2010sucking ; cohen2011active ; sarabadani2014iso ; sarabadani2015theory
; sarabadani2017driven ; sarabadani2018theory ; sarabadani2018dynamics ;
menais2018polymer ; sarabadani2020pulling ; dubbeldam2007polymer ;
gauthier2009nondriven ; huopaniemi2006langevin ; luo2007influence ;
luo2008dynamical ; milchev2004polymer ; palyulin2014polymer , confined in one
dimension luo2010polymerpre ; luo2010polymerjcp ; sarabadani2022driven , or
confined in two dimensions ali2006polymer ; das2019dynamics ;
goldfeld2009packaging ; purohit2003mechanics ; sakaue2007semiflexible ;
smith2001bacteriophage ; sun2018theoretical ; zhang2014polymer . The latter
case corresponds to the two dimensional model for packing of a DNA or RNA of a
virus inside a capsid.

Figure 1: Schematic of an active polymer translocation through a nano-pore
into a rigid two dimensional circular cavity with radius of $R=20$, (a) at
$t=0$ (beginning of the translocation process), and (b) at $t=\tau$ (end of
the process). The rigid wall and two dimensional circular rigid cavity, which
are depicted by green beads, are located in the $x-y$ plane. The wall, which
is parallel to the $y$ axis, is located at $x=0$. The circular cavity whose
center is at $(R,0)$ has been tangentially attached to the wall. The center of
the nano-pore with width of $D=1.5$ is located at $(0,0)$. Magnified picture
shows the tail (in blue) and head (in red) monomers belonging to a segment. At
each time step the SP force acts on the red head monomer in the direction from
the tail to the head bead. The contour length of the polymer is $N=200$ and
its persistence length is $l_{p}=5$. For this figure the magnitude of the SP
force is $F_{\textrm{sp}}=2$.

To pass through a nano-pore the conformation of a polymer has to be changed
and due to decrease in the available polymer conformations during the
translocation process, the polymer feels an entropic barrier. To have a
successful translocation event, the polymer must overcome this entropic
barrier by using different strategies such as external fields
sigalov2008detection ; meller2001voltage , flow-induced auger2014zero ;
chen2021dynamics ; ding2021flow ; sakaue2005flow ; zheng2018revisiting ,
chaperon-assisted khalilian2021polymer ; abdolvahab2011sequence ;
ambjornsson2005directed ; chen2013dynamics ; emamyari2017polymer ;
gopinathan2007polymer ; taheri2018granular ; yu2011chaperone ; zandi2003drives
, and pulling techniques like atomic force microscope (AFM), optical and
magnetic tweezers ritort2006single ; keyser2006direct . From a theoretical
point of view, driven translocation of a polymer through a nano-pore can be
divided into two categories pore-driven bhattacharya2009scaling ;
luo2008dynamics ; rowghanian2011force ; saito2011dynamical ; saito2012process
; sakaue2007nonequilibrium ; sakaue2010sucking ; sakaue2016dynamics ;
sarabadani2014iso ; sarabadani2015theory ; sarabadani2018theory ;
sarabadani2017driven ; cohen2011active ; zhang2014polymer  and end-pulled
menais2018polymer ; huopaniemi2007polymer ; menais2016polymer ;
sarabadani2018dynamics ; sarabadani2018theory ; sarabadani2020pulling . In the
present paper, instead of the explicit external driving force, a new strategy
is considered in which the self-propulsion of the polymer plays an effective
driving force role that facilitates the translocation.

Active polymers and filamentous structures play a crucial role in the
biological systems and have various experimental applications
phillips2012physical ; howard2001_book . Accordingly, dynamical and
conformational properties of active polymers have been studied theoretically
(through both the analytical as well as the simulation models) anand2018_pre
; Gompper_tangentional_activity ; eisenstecken2017_jcp ; eisenstecken2022_jcp
; isele2015_sm ; laskar2017_njp ; peterson2020_JSM-TE ; sarkar2016_pre  and
experimentally dreyfus2005_Nature ; hill2014_ACS-Nano ; nishiguchi2018_njp ;
schaller2010_Nature . As a manifestation of active polymer applications, the
large group of biological and artificial micro-swimmers, by using filamentous
and hairy like organelles, such as flagella and cilia as the active bio-
polymers, produce a net locomotion in the low Reynolds regimes
phillips2012physical ; howard2001_book ; brennen1977_ARFM ; elgeti2015_RPP ;
bechinger2016_RMP .

In this paper, our purpose is to study the translocation of an active polymer
through a nano-pore into a two dimensional circular cavity by overcome the
entropic barrier. The present paper is organized as follows: In Sec. II the
Langevin dynamics (LD) simulation method is explained in detail and it is
introduced how to model a coarse grained self-propelled semi-flexible polymer
as well as a rigid two dimensional cavity. Then in Sec. III the results are
presented. Finally, summary and conclusions are discussed in Sec. IV.

##  II Models and Methods

###  II.1 Simulation model

This paper is devoted to present our study on the translocation of a self-
propelled semi-flexible polymer, consisting of $N$ identical monomers, through
a nano-pore into a two dimensional circular cavity using the Langevin dynamics
(LD) simulation method by means of the LAMMPS package LAMMPS . The
configurations of the system at the beginning of the translocation process at
time $t=0$, and at the end of the translocation process at time $t=\tau$
($\tau$ is the translocation time) have been illustrated in Fig. 1(a) and
(b), respectively. As shown a rigid wall, which is located in the $x-y$ plane
parallel to the $y$ axis at $x=0$, separates the system into cis and trans
sides. The center of the nano-pore, whose width is $D=1.5$, is located at
$(0,0)$ and is at the intersection of the wall and cavity boarder. Moreover,
the center of a rigid cavity with radius of $R$ is at $(R,0)$ and the cavity
has been attached tangentially to the wall. It should be mentioned that the
wall and the cavity have been made by similar beads with size $\sigma$. During
the translocation process a self-propelled polymer translocates through a
nano-scale pore from the cis side ($x<0$), into the cavity located in the
trans side ($x>0$).

###  II.2 Langevin dynamics simulation

In our simulations, a semi-flexible self-avoiding polymer composed of $N$
monomers has been studied as a bead-spring chain. The bonded interactions are
the sum of the Weeks-Chandler-Anderson (WCA) and finitely extensible nonlinear
elastic (FENE) potentials. The repulsive WCA potential is

$U_{\textrm{WCA}}(r)=\begin{cases}U_{\textrm{LJ}}(r)-U_{\textrm{LJ}}(r_{\textrm{c}}),&\text{if
$r\leq r_{\textrm{c}}$};\\\ 0,&\text{if $r>r_{\textrm{c}}$,}\end{cases}$ (1)

where $r$ is the distance between two monomers, $r_{\textrm{c}}=2^{1/6}$ is
the cut-off radius and
$U_{\textrm{LJ}}(r)=4\varepsilon\big{[}\big{(}\sigma/r\big{)}^{12}-\big{(}\sigma/r\big{)}^{6}\big{]}$
is the Lennard-Jones (LJ) potential with $\varepsilon$ and $\sigma$ as the
depth of the potential and the monomer size, respectively. The consecutive
bonded monomers are connected to each other by the FENE potential

$U_{\textrm{FENE}}(r)=-\frac{1}{2}kR^{2}_{0}\ln\bigg{[}1-\bigg{(}\frac{r}{R_{0}}\bigg{)}^{2}\bigg{]},$
(2)

where $R_{0}$ and $k$ are the maximum allowed distance between the monomers
and the spring constant, respectively. The stiffness of the chain comes from
the bending potential

$U_{\textrm{b}}(\theta)=k_{\textrm{b}}[1+\cos(\theta)],$ (3)

where $k_{\textrm{b}}$ and $\theta$ denote the bending energy coefficient and
angle between two consecutive bonds, respectively. In the two dimensional
system, the persistence length is given by
$l_{\textrm{p}}=2k_{\textrm{b}}/(k_{\textrm{B}}T)$, where $k_{\textrm{B}}$ and
$T$ are the Boltzmann constant and temperature, respectively. Moreover, the
non-bonded interactions between monomers and the wall as well as the cavity
are WCA.

To activate the polymer, the contour length of the polymer is divided to
$n=N/l_{\textrm{p}}$ segments each with the length of $l_{\textrm{p}}$. In the
system under consideration, the contour length and the persistence length of
the polymer are constant and are $N=200$ and $l_{\textrm{p}}=5$, respectively.
Therefore, the polymer consists of $n=40$ segments. As can be seen from the
magnified segment of the polymer in Fig. 1(a), the tail and head monomers are
in blue and red colors, respectively, and three beads in the body of each
segment are in black color. At each time step, the direction of the self-
propelling (SP) force on each segment is updated according to the updated
positions of the monomers, and a SP force $F_{\textrm{sp}}$ acts on the head
of each segment in the direction of the tail bead toward the head bead of that
segment.

As mentioned above the LD simulation method has been employed to simulate the
dynamics of the system. Therefore, the equation of motion of the $i$th monomer
is written as

$M\ddot{\vec{r}}_{i}=-\vec{\nabla}U_{i}-\eta\dot{\vec{r}}_{i}+F_{\textrm{sp}}\delta_{i\textrm{h}}\mathbb{\hat{e}}_{\textrm{h}}+\vec{\xi}_{i}(t),$
(4)

where, $\vec{r}_{i}$ is the position vector of the $i$th monomer and $U_{i}$
is the sum of all interaction potentials experienced by the $i$th monomer. The
$M$, $\eta$ and $F_{\textrm{sp}}$ denote the monomer mass, friction
coefficient of the solvent, and the magnitude of the SP force, respectively.
$\delta_{i\textrm{h}}$ is the Kronecker delta function for ensuring that the
SP force $F_{\textrm{sp}}$ just acts on the head of each segment and
$\mathbb{\hat{e}}_{\textrm{h}}$ represents the unit vector at the direction of
the tail monomer to the head monomer in each segment at the corresponding
time. $\vec{\xi}_{i}(t)$ is the thermal white noise at time $t$ with the
average of $\langle\vec{\xi}_{i}(t)\rangle=\vec{0}$ and
$\langle\vec{\xi}_{i}(t)\cdot\vec{\xi}_{j}(t^{\prime})\rangle=4\eta
k_{\textrm{B}}T\delta_{ij}\delta(t-t^{\prime})$, in which $\delta_{ij}$ and
$\delta(t-t^{\prime})$ are Kronecker and Dirac delta functions, respectively.

The $M$, $\sigma$ and $\varepsilon$ have been used as the units of mass,
length and energy, respectively. The mass of each bead (including the monomers
and also wall and cavity beads) is $M=1$ and the value of its diameter is
$\sigma=1$. In addition, the values of other parameters are $\varepsilon=1$,
$\eta=0.7$, $k_{\textrm{B}}T=1.2$, $k=30$ and $R_{0}=1.5$, and the value of
the bending energy coefficient has been chosen as $k_{\textrm{b}}=3$ which is
equivalent to the persistence length of $l_{p}=5$ in two dimension. The size
of the simulation box in the $x$ and $y$ directions are $L_{x}=L_{y}=420$ and
there is a periodic boundary condition in the $y$ direction on the cis side.

The initial configuration of the system is the straight polymer chain that its
first two monomers are inside the cavity while its third monomer is fixed at
the center of the nano-pore and the rest of the polymer beads are in the cis
side. The LD simulations contain two stages. In the first stage, while the
third polymer bead is fixed at the nano-pore the polymer is well equilibrated
during the time interval of $t_{\textrm{eq}}=2\times 10^{4}$ (in LJ unites).
During the equilibration stage the polymer is passive. After the equilibration
stage the fixed monomer at the nano-pore is released and simultaneously the SP
forces are switched on and act on the heads of segments, and the second stage
that is the actual translocation process of the polymer into the cavity is
started.

##  III Results

In this section, the results of the translocation of a semi-flexible polymer
composed of $N=200$ monomers, with persistence length $l_{\textrm{p}}=5$, into
a circular cavity with various values of radius $R=10$, $12$, $15$, $20$,
$30$, $40$, and $50$ are considered. The chosen values for the SP force are
$F_{\textrm{sp}}=1.2$, 2, 5, 10, and 15. To find the mean values of different
quantities, the averages have been performed over 1000 uncorrelated
trajectories.

###  III.1 Translocation time

Figure 2: (a) The average of the translocation time $\langle\tau\rangle$ as a
function of the cavity radius $R$, for different values of the SP force
$F_{\textrm{sp}}=1.2$ (black circles), 2 (red squares), 5 (green diamonds), 10
(blue triangles up) and 15 (orange triangles left). The inset shows the
normalized average translocation time
$\langle\tau\rangle/\langle\tau(R=10)\rangle$ as a function of $R$. (b) The
$\langle\tau\rangle$ as a function of the SP force $F_{\textrm{sp}}$ for
various values of $R=10$ (black circles), 12 (red squares), 15 (green
diamonds), 20 (blue triangles up), 30 (orange triangles left), 40 (cyan
triangles down) and 50 (violet triangles right). The black dashed and violet
dashed-dotted lines are fit to the data with $R=10$ and 50, respectively, but
shifted for better visibility. The inset presents the force exponent $\beta$
of the scaling form $\langle\tau\rangle\propto F_{\textrm{sp}}^{\beta}$ as a
function of $R$.

The most important quantity, that characterizes the translocation dynamics
both in the theory and experiment, is the translocation time $\tau$. The
$\tau$ is the time that takes for the whole polymer to translocate into the
trans side, i.e. into the cavity for the present system under consideration
(see Fig. 1). In Fig. 2(a) the average translocation time
$\langle\tau\rangle$ has been plotted as a function of the cavity radius $R$,
for various values of the SP force $F_{\textrm{sp}}=1.2$ (black circles), 2
(red squares), 5 (green diamonds), 10 (blue triangles up), and 15 (orange
triangles left). In the inset the normalized average translocation time
$\langle\tau\rangle/\langle\tau(R=10)\rangle$ has been plotted as a function
of $R$. As can be seen $\langle\tau\rangle$ decreases monotonically as $R$
increases. Inset shows that the decreasing of $\langle\tau\rangle$ is more
pronounced at weak force limit. Panel (b) presents $\langle\tau\rangle$ as a
function of $F_{\textrm{sp}}$ for different values of the cavity radius $R=10$
(black circles), 12 (red squares), 15 (green diamonds), 20 (blue triangles
up), 30 (orange triangles left), 40 (cyan triangles down) and 50 (violet
triangles right). The black dashed and violet dashed-dotted lines are fitting
to the data for $R=10$ and 50, respectively, but shifted for better
visibility. The inset in panel (b) shows the force exponent $\beta$, that is
defined as $\langle\tau\rangle\propto F_{\textrm{sp}}^{\beta}$, as a function
of the circular cavity radius $R$. As can be seen the value of $\beta$
approaches $-1$ as long as the radius of the circular cavity is much less than
gyration radius of the passive semi-flexible polymer in free space in two
dimensions $R_{\textrm{g}}\approx 23$, i.e. $R\ll R_{\textrm{g}}$. Then,
$\beta$ increases rapidly by increasing the value of $R$ in the region $R\leq
R_{\textrm{g}}$ and in the region $R\geq R_{\textrm{g}}$, it takes its
asymptotic value $\beta\approx-0.92$.

Figure 3: (a) Probability distribution function of the translocation times
$P(\tau)$ as a function of translocation time $\tau$, for fixed value of the
cavity radius $R=10$ and various values of the SP force $F_{\textrm{sp}}=15$
(orange bars), 10 (blue bars), 5 (green bars), 2 (red bars), and 1.2 (black
bars). Panels (b), (c), (d) and (e) are the same as panel (a) but for
different values of $R=20$, 30, 40 and 50, respectively. (f) The $P(\tau)$ as
a function of $\tau$, for fixed value of the SP force $F_{\textrm{sp}}=1.2$
and various values of $R=50$ (orange bars), 40 (blue bars), 30 (green bars),
20 (red bars) and 10 (black bars). Panels (g), (h), (i) and (j) are the same
as panel (f) but for different values of $F_{\textrm{sp}}=2$, 5, 10 and 15,
respectively. (k) Normalized value of the standard deviation
$\sigma_{\textrm{s}}/\langle\tau\rangle$ as a function of $R$, for various
values of $F_{\textrm{sp}}=1.2$ (black circles), 2 (red squares), 5 (green
diamonds), 10 (blue triangles up) and 15 (orange triangles left). (l) The
$\sigma_{\textrm{s}}/\langle\tau\rangle$ as a function of $F_{\textrm{sp}}$
for different values of $R=10$ (black circles), 12 (red squares), 15 (green
diamonds), 20 (blue triangles up), 30 (orange triangles left), 40 (cyan
triangles down) and 50 (violet triangles right).

Figure 4: (a) Waiting time $w$ as a function of the translocation coordinates
$s$ for a fixed value of the cavity radius $R=10$ and various values of the SP
force $F_{\textrm{sp}}=1.2$ (black line), 2 (red line), 5 (green line), 10
(blue line) and 15 (orange line). Panels (b), (c) and (d) are the same as
panel (a) but for different values of $R=12$, 20 and 50, respectively. (e) The
$w$ as a function of $s$ for a fixed value of $F_{\textrm{sp}}=1.2$ and
various values of $R=10$ (black line), 12 (red line), 20 (green line) and 50
(blue line), respectively. Panels (f), (g) and (h) are the same as panel (e)
but for different values of $F_{\textrm{sp}}=2$, 5 and 10, respectively.

In Fig. 3(a) the probability distribution function of translocation times
$P(\tau)$ has been plotted as a function of translocation time $\tau$ for
different values of the SP force $F_{\textrm{sp}}=1.2$ (black bars), 2 (red
bars), 5 (green bars), 10 (blue bars), and 15 (orange bars), and fixed value
of the cavity radius $R=10$. Panels (b), (c), (d) and (e) are the same as
panel (a) but for cavity radii of $R=20$, 30, 40 and 50, respectively. As can
be seen in each panel, for a given value of $R$, as the value of the SP force
decreases the average value of translocation time $\langle\tau\rangle$
increases and the histograms get wider (note that the horizontal axis is in
logarithmic scale). To illustrate the width of $P(\tau)$s of panels (a)–(e),
in panel (k) the normalized standard deviation
$\sigma_{\textrm{s}}/\langle\tau\rangle$ has been plotted as a function of the
radius of the circular cavity $R$, for various values of the SP force
$F_{\textrm{sp}}=1.2$ (black circles), 2 (red squares), 5 (green diamonds), 10
(blue triangles up), and 15 (orange triangles left). According to panels (a)
to (e), $\sigma_{\textrm{s}}/\langle\tau\rangle$ decreases monotonically by
increasing the value of $R$ and then for $R\gtrsim R_{\textrm{g}}$ it takes
its asymptotic constant value for each value of the SP force. Panel (f) shows
$P(\tau)$ as a function of $\tau$ for different values of cavity radius $R=10$
(black bars), 20 (red bars), 30 (green bars), 40 (blue bars), and 50 (orange
bars), for fixed value of SP force $F_{\textrm{sp}}=1.2$. Panels (g), (h), (i)
and (j) are the same as panel (f) but for different values of SP force
$F_{\textrm{sp}}=2$, 5, 10 and 15, respectively. As seen, for a given value of
$F_{\textrm{sp}}$ the probability distribution function of a translocation
time $P(\tau)$ doesn’t change significantly for $R\gtrsim R_{\textrm{g}}$.
Panel (l) shows the same value as in panel (k) but as a function of the SP
force, for different values of the radius of the cavity $R=10$ (black
circles), 12 (red squares), 15 (green diamonds), 20 (blue triangles up), 30
(orange triangles left), 40 (cyan triangles down), 50 (violet triangles
right). It can be seen that $\sigma_{\textrm{s}}/\langle\tau\rangle$
monotonically decreases by increasing the SP force, and for $R\gtrsim
R_{\textrm{g}}$ the curves collapse on a master curve. In summary, for a given
value of $R$, the distributions of translocation times (and also the mean
translocation times) significantly deviate from each other by increasing the
value of $F_{\textrm{sp}}$ [see panels (a)–(e)]. On the other hand, for a
given value of $F_{\textrm{sp}}$, the distributions of translocation times do
not separate by changing the value of $R$ for the region $R\gtrsim
R_{\textrm{g}}$ [see panels (f)–(j)]. This results are in agreement with
panels (a) and (b) in Fig. 2.

###  III.2 Waiting time distribution

Figure 5: (a) $x$ component of the monomer velocity $v_{x}$ as a function of
the translocation coordinates $s$ for fixed values of $R=10$ and
$F_{\textrm{sp}}=1.2$ at different times $t/\langle\tau\rangle=0.1$ (red
line), 0.3 (blue line), 0.5 (brown line) and 0.8 (cyan line). Panels (b), (c),
(d) and (e) are the same as panel (a), but for different values of the SP
force $F_{\textrm{sp}}=2$, 5, 10 and 15, respectively. Panels (f) to (j), and
panels (k) to (o), are the same as panels (a) to (e), but for different values
of $R=20$ and 50, respectively. Empty circles show mean value of the index of
monomer inside the pore at the corresponding times.

Waiting time (WT) $w$, which is the time that each monomer spends inside the
nano-pore during the translocation process, is an important quantity that
reveals the dynamics of the translocation process at the monomer level. In
Fig. 4(a) the WT has been shown as a function of the translocation coordinate
$s$ for fixed values of the cavity radius $R=10$ and various values of the SP
force $F_{\textrm{sp}}=1.2$ (black line), 2 (red line), 5 (green line), 10
(blue line), and 15 (orange line). Panels (b), (c), and (d), are the same as
panel (a), but for different values of the cavity radius $R=12$, 20, and 50,
respectively. The translocation coordinate $s$ is the number of monomers at
the trans side. The values of $s=1$ and $s=200$ correspond to the first
monomer inside the cavity and the last monomer on the other side of the
polymer, respectively. It can be seen that in all panels for the fixed value
of $R$ at a given value of $s$, the value of the WT decreases as
$F_{\textrm{sp}}$ increases. As mentioned above, during the equilibration
process, the first and the second monomers of the polymer corresponding to
$s=1$ and $s=2$, respectively, are inside the cavity and the third monomer
with $s=3$ is fixed at the nano-pore. At the beginning of the actual
translocation process, when the third monomer is released and the SP force is
applied, the first and second monomers have back and forth motions, specially
in the weak force limit. Therefore, the value of the WT for $s=1$ and 2 is not
zero and it has a small value (not shown in Fig. 4). The WT has a local
maximum at $s=3$, then by increasing $s$ the total net force acting on the
trans-side subchain increases, therefore the value of $w$ decreases. At the
end of translocation process $w$ increases as the excluded volume effect grows
and resists against the translocation of the monomers into the trans side.
This effect is more pronounced for small values of cavity radius $R=10$ and
weak force limit $F_{\textrm{sp}}\lesssim 2$.

Figure 4(e) shows the WT in terms of $s$ for a fixed value of SP force
$F_{\textrm{sp}}=1.2$ and various values of $R=10$ (black line), 12 (red
line), 20 (green line), and 50 (blue line). Panels (f), (g) and (h) are the
same as panel (e), but for $F_{\textrm{sp}}=2$, 5, and 10, respectively. As
the value of the SP force increases [from panel (e) to panel (h)] the curves
for different values of the $R$ get closer to each other, and for
$F_{\textrm{sp}}=10$ in panel (h) they are almost coincide each other.
Consequently, for weak force limit, e.g. for $F_{\textrm{sp}}=1.2$ in panel
(e) the value of $R$ plays the dominant role in the dynamics of the
translocation process, while in strong force limit, e.g. $F_{\textrm{sp}}=10$
in panel (h), just the SP force controls the translocation dynamics.

###  III.3 Monomers’ velocities distribution

The monomers’ velocities in $x$ direction (from cis to trans) $v_{x}$ at time
$t$, which is calculated by obtaining the average velocity of each monomer in
a narrow time window of $[t-\delta t/2,t+\delta t/2]$, where $\delta t=1$,
shows the dynamics of the movement of monomers at different times during the
translocation process. In Fig. 5(a) $v_{x}$ has been plotted as a function of
the translocation coordinates $s$ for fixed values of the cavity radius $R=10$
and the SP force $F_{\textrm{sp}}=1.2$, and at various times
$t/\langle\tau\rangle=0.1$ (red line), 0.3 (blue line), 0.5 (brown line) and
0.8 (cyan line). Panels (b), (c), (d) and (e) are the same as panel (a) but
for different values of the SP force $F_{\textrm{sp}}=2$, 5, 10 and 15,
respectively. Panels (f)–(j), and panels (k)–(o) are the same as panels
(a)–(e), but for different values of the cavity radius $R=20$ and 50,
respectively. Empty circles show mean value of the index of the monomer inside
the nano-pore at the corresponding times. These plots indicate that $v_{x}$
exhibits oscillatory pattern for those monomers inside the cavity. At constant
values of $R$ and $F_{\textrm{sp}}$ with the passage of time, more monomers
involved in the oscillatory behavior. Moreover, at constant $R$ as the
magnitude of $F_{\textrm{sp}}$ increases [from left to right in each row, e.g.
from (a) to (e)], the magnitude of the pattern in the oscillatory part of the
curves grows. In addition, at constant value of $F_{\textrm{sp}}$ as the value
of $R$ increases [from top to bottom in each column, e.g. panels (e)–(j)–(o)]
the number of periods decreases, and it is more pronounced at longer times.
The oscillations comes from the fact that the polymer subchain in the trans
side (inside the cavity) is folded. If the value of cavity radius is chosen
very small, e.g. $R=10$ [panels (a) to (e)], then a spiral is constructed
inside the cavity. For constant value of $R=10$, at the long time limit, i.e.
$t=0.8\langle\tau\rangle$, the oscillations in $v_{x}$ curve has a damping
behavior for those monomers that have entered the cavity first (with small
values of $s$) and are located at the inner loops in the spiral due to the
lack of enough space. In addition, the monomers that have translocated earlier
(with smaller values of $s$) experience weaker net effective force compared to
those upcoming monomers at the later times with larger values of $s$.
Therefore, the monomers with small values of $s$ have slower dynamics than the
monomers with larger values of $s$. This leads to the larger values of the
amplitude of $v_{x}$ for those monomers located at outer layers in the spiral
with larger values of $s$ compared to those located at the inner layers with
smaller values of $s$. As will be discussed in Sub-sec. III.4, the strong SP
force leads to a relatively perfect spiral inside the cavity with radius
$R=10$, while an irregular spiral conformation is made by applying a weak SP
force. Another important feature exhibited in Fig. 5 is that as the value of
$R$ increases (from first row in the top to the last row in the bottom) the
$v_{x}$ gets more positive values due to the less excluded volume
interactions.

Figure 6:  (a) Average of the magnitude of the turning number
$\langle|\psi|\rangle$ as a function of the normalized time
$t/\langle\tau\rangle$ for fixed value of the SP force $F_{\textrm{sp}}=1.2$
and various values of $R=10$ (black circles), 20 (blue triangles up), 30
(orange triangles left), 40 (cyan triangles down) and 50 (violet triangles
right). Panels (b), (c) and (d) are the same as panel (a) but for different
values of $F_{\textrm{sp}}=5$, 10 and 15, respectively.

###  III.4 Turning number

To quantify the properties of the active polymer configurations inside the
cavity, the quantity $\psi$, which is the turning number (TN), is introduced
as krantz1999handbook

$\psi=\frac{1}{2\pi}\sum_{i=1}^{n}\left[\theta_{i+1}-\theta_{i}\right],$ (5)

where $i$ and $n$ are the bond index and the total number of bonds inside the
cavity at time $t$, respectively, and $\theta_{i}$ is the angle between the
$x$ axis and the unit vector $\hat{t_{i}}$ of the bond with index $i$, that is
defined by $\hat{t_{i}}=\left(\cos\theta_{i},\sin\theta_{i}\right)$.

To study the time evolution of the layered structure of the folded polymer
sub-chain inside the cavity, in Fig. 6(a) the average of the magnitude of the
turning number $\langle|\psi|\rangle$ has been plotted as a function of
normalized time $t/\langle\tau\rangle$ for fixed value of SP force
$F_{\textrm{sp}}=1.2$ and various values of cavity radius $R=10$ (black
circles), 20 (blue triangles up), 30 (orange triangles left), 40 (cyan
triangles down) and 50 (violet triangles right). Panels (b), (c) and (d) are
the same as panel (a) but for different values of $F_{\textrm{sp}}=5$, 10 and
15, respectively. For fixed value of SP force in each panel, as the value of
$R$ increases the value of $\langle|\psi|\rangle$ at a given time decreases
and approaches to a lower limit value at that moment. Indeed, in each panel at
constant value of $F_{\textrm{sp}}$ for $R\gg R_{\textrm{g}}$ the curves
almost coincide each other. On the other hand for smallest value of the cavity
radius $R=10$ and at a constant value of time as the SP force increases [black
circles from panel (a) to panel (d)] the value of the $\langle|\psi|\rangle$
increases, that means as the SP force gets stronger a more regular spiral is
constructed. In contrast, for $R>R_{\textrm{g}}$ the value of the turning
number at a certain time gradually decreases by increasing the value of the SP
force.

Figure 7: (a) The magnitude of the turning number $\langle|\psi|\rangle$ for
the final snapshot of the system at $t=\tau$, as function of the SP force
$F_{\textrm{sp}}$ for various values of the cavity radius $R=10$ (black
circles), 12 (red squares), 15 (green diamonds), 20 (blue triangles up), 30
(orange triangles left), 40 (cyan triangles down) and 50 (violet triangles
right). (b) The $\langle|\psi|\rangle$ as function of $R$ at $t=\tau$, for
different values of $F_{\textrm{sp}}=15$ (orange triangles left), 10 (blue
triangles up), 5 (green diamonds), 2 (red squares) and 1.2 (black circles).
(c) Phase diagram of $\langle|\psi|\rangle$ at $t=\tau$ in
$R$–$F_{\textrm{sp}}$ plane. The colors cyan, blue, orange and red indicate
the values of $\langle|\psi|\rangle<1$, $1<\langle|\psi|\rangle<2$,
$2<\langle|\psi|\rangle<3$ and $\langle|\psi|\rangle>3$, respectively.

Figure 8: (a) Probability distribution function of the turning number
$P(\psi)$ as a function of turning number $\psi$ for the final snapshot of the
system at $t=\tau$, for fixed value of cavity radius $R=10$ and various values
of the SP force $F_{\textrm{sp}}=15$ (orange diamonds), 5 (green squares) and
1.2 (black circles). Panels (b) and (c) are the same as panel (a) but for
different values of $R=20$ and 50, respectively. (d) The $P(\psi)$ as a
function of $\psi$ at $t=\tau$, for fixed value of $F_{\textrm{sp}}=1.2$ and
various values of $R=10$ (black circles), 20 (blue squares) and 50 (violet
diamonds). Panels (e) and (f) are the same as panel (d) but for different
values of SP force $F_{\textrm{sp}}=5$ and 15, respectively.

Figure 9:  (a) Monomer number density $\rho$ for fixed values of cavity radius
$R=10$ and SP force $F_{\textrm{sp}}=1.2$ at time $t=0$. Panels (b), (c), (d),
(e) and (f) are the same as panel (a) but at different times
$t/\langle\tau\rangle=0.2$, 0.4, 0.6, 0.8 and the last snapshot, respectively.
Panels (g)–(l) are the same as panels (a)–(f), respectively, but for
$F_{\textrm{sp}}=15$.

The value of $\langle|\psi|\rangle$ at the final snapshot of the simulation,
when the translocation process terminates, gives information about the
structure of the polymer inside the cavity at $t=\tau$. To investigate the
polymer structure for the final configuration of the polymer inside the
cavity, in Fig. 7(a) $\langle|\psi|\rangle$ has been plotted as a function of
$F_{\textrm{sp}}$ for various values of $R=10$ (black circles), 12 (red
squares), 15 (green diamonds), 20 (blue triangles up), 30 (orange triangles
left), 40 (cyan triangles down) and 50 (violet triangles right). As the value
of $F_{\textrm{sp}}$ increases the value of $\langle|\psi|\rangle$ grows
monotonically for $R\lesssim R_{\textrm{g}}$, where $R_{\textrm{g}}\approx
23$. On the other hand, for $R>R_{\textrm{g}}$, with the increase in the SP
force, first the value of $\langle|\psi|\rangle$ grows, then gets its maximum
and after that it decreases.

Panel (b) in Fig. 7 shows $\langle|\psi|\rangle$ as a function of $R$ but for
various values of SP force $F_{\textrm{sp}}=1.2$ (black circles), 2 (red
squares), 5 (green diamonds), 10 (blue triangles up) and 15 (orange triangles
left). As seen in the region $R\lesssim R_{\textrm{g}}$ and at constant $R$ as
value of the SP decreases the value of $\langle|\psi|\rangle$ decrease too. On
the other hand, in the region $R>R_{\textrm{g}}$ and at constant $R$, for
$F_{\textrm{sp}}\geq 5$ as the value of the SP force increases the value of
$\langle|\psi|\rangle$ decreases, and for $F_{\textrm{sp}}\leq 2$ it’s vice
versa.

In panel (c) in Fig. 7 the results of panels (a) and (b) have been
qualitatively summarized by drawing a phase diagram for quantity
$\langle|\psi|\rangle$ for the final snapshot of the polymer configuration at
$t=\tau$ in the phase space $R$–$F_{\textrm{sp}}$. According to the values of
$\langle|\psi|\rangle$ for different sets of $R$–$F_{\textrm{sp}}$, the phase
space has been divided into four regions $\langle|\psi|\rangle<1$ (in cyan
color), $1<\langle|\psi|\rangle<2$ (in blue color), $2<\langle|\psi|\rangle<3$
(in orange color) and $\langle|\psi|\rangle>3$ (in red color).

To investigate the turning number $\psi$ in more detail, in Fig. 8(a) the
probability distribution function of the turning number $P(\psi)$ has been
plotted in terms of $\psi$ at $t=\tau$ for the whole active polymer inside the
cavity for fixed value of $R=10$ and various values of the SP force
$F_{\textrm{sp}}=1.2$ (black circles), 5 (green squares) and 15 (orange
diamonds). Panels (b) and (c) are the same as panel (a) but for different
values of $R=20$ and 50, respectively. Panel (d) shows $P(\psi)$ as a function
of $\psi$ for fixed value of $F_{\textrm{sp}}=1.2$ and various values of
cavity radius $R=10$ (black circles), 20 (blue squares) and 50 (violet
diamonds). Panels (e) and (f) are the same as panel (d), but for different
values of $F_{\textrm{sp}}=5$ and 15, respectively.

According to the horizontal axis of the panels in Fig. 8, the value of $\psi$
can be negative (positive) that shows the clockwise (counter clockwise)
conformation of a spiral. As can be seen from panel (a), the translocation of
an active polymer for $F_{\textrm{sp}}=15$ into a cavity with radius $R=10$,
leads to two peaks at $\psi\approx-5$ and 5 in $P(\psi)$. In the region
between the two peaks the value of $P(\psi)$ are very small compared to its
values at the peaks. Infact for the smallest chosen value of the cavity radius
$R=10$ and strong SP force $F_{\textrm{sp}}=15$, the active polymer folds to
make mostly a conformation with regular spiral structure with $\psi\approx-5$
($+5$) as a clockwise (counter clockwise). For smaller values of the SP force,
the height of the peaks decreases, and instead a hump in the region between
the side peaks appears with a maximum around $\psi=0$. Indeed, for $R=10$ by
decreasing the value of SP force the translocation process takes place in such
a way that the polymer folds inside the cavity and finally at $t=\tau$ gets an
irregular spiral conformation. This result is in agreement with the results of
previous subsections.

Figure 10:  (a) Monomer number density $\rho$ for fixed values of cavity
radius $R=20$ and SP force $F_{\textrm{sp}}=1.2$ at time $t=0$. Panels (b),
(c), (d), (e) and (f) are the same as panel (a) but at different times
$t/\langle\tau\rangle=0.2$, 0.4, 0.6, 0.8 and the last snapshot, respectively.
Panels (g)–(l) are the same as panels (a)–(f), respectively, but for
$F_{\textrm{sp}}=15$.

Figure 11:  (a) Monomer number density $\rho$ for fixed values of cavity
radius $R=50$ and SP force $F_{\textrm{sp}}=1.2$ at time $t=0$. Panels (b),
(c), (d), (e) and (f) are the same as panel (a) but at different times
$t/\langle\tau\rangle=0.2$, 0.4, 0.6, 0.8 and the last snapshot, respectively.
Panels (g)–(l) are the same as panels (a)–(f), respectively, but for
$F_{\textrm{sp}}=15$.

For the strong SP force $F_{\textrm{sp}}=15$ (orange diamonds) as the value of
$R$ increases from panel (a) to (c) the height of the side peaks in the
distribution decreases and their locations get closer to each other. and
finally in panel (c) the two side peaks are merged. Moreover, for smaller
values of the SP force by increasing the value of $R$ the height of the side
peaks gets smaller and the hump at the middle of the distribution is more
pronounced [from panel (a) to (c)] and the dominant part of the distribution
is in the middle hump.

Indeed, for $R=50$ in panel (c) the decrease in the value of the SP force
leads to a relatively wider distribution. This confirms that by increasing the
size of the circular cavity, the twine picks and therefore the regular spiral
structure of the active polymer inside the cavity disappear [see panels (b)
and (c) in Fig. 7].

###  III.5 Monomer density

The time evolution of the spatial configuration of the active polymer during
the translocation process can be considered by the monomers number density
$\rho$. To quantify the $\rho$, the simulation box is divided into square
cells each with size of $1\times 1$. Then the average numbers of monomers that
occupy the cells are obtained in a narrow window of time $\Delta t=1$ around a
desired moment for each trajectories and after that the average is taken over
all trajectories at the corresponding time as well as over the narrow window
of time. The density for the final time step of the translocation process is
obtained only by taking the average over the trajectories.

Fig. 9 demonstrates the time evolution of $\rho$. Panel (a) shows $\rho$ just
at the beginning of the translocation process at $t=0$ for fixed values of
$R=10$ and $F_{\textrm{sp}}=1.2$. Panels (b), (c), (d), (e) and (f) are the
same as panel (a) but at times $t=0.2\langle\tau\rangle$,
$0.4\langle\tau\rangle$, $0.6\langle\tau\rangle$, $0.8\langle\tau\rangle$ and
at the final time step of the translocation process, respectively. As seen in
panel (a) at time $t=0$ the monomers are symmetrically distributed around the
horizontal axis passing through the center of the nano-pore. Panels (b) to (f)
show that for small cavity with $R=10$ at a weak force limit
$F_{\textrm{sp}}=1.2$ the monomers are almost uniformly distributed inside the
cavity during the translocation process. Panels (g)–(l) are the same as panels
(a)–(f) but for strong value of $F_{\textrm{sp}}=15$. Panel (h), which
presents $\rho$ at $t=0.2\langle\tau\rangle$, clearly shows that for strong
value of SP force the dynamics is very fast and the sub-chain inside the
cavity does not have enough time to fold itself. Therefore, the polymer is
able to touch the cavity wall in front of the nano-pore in the other side of
the cavity at short time limit. With the passage of the time, as presented in
panels (i) to (l), the sub-chain inside the cavity folds due to the
interaction with the cavity wall. This leads to a ring structure in vicinity
of the cavity wall as seen at $t=0.4\langle\tau\rangle$ in panel (i) and at
the later times a regular structure which has a spiral configuration is being
made as illustrated in panels (j) to (l).

In Fig. 9, the results of panels (a)–(f) and panels (g)–(l) for weak and
strong SP force limits, respectively, are in agreement with the results shown
in Fig. 8(a). Indeed for weak force limit of $F_{\textrm{sp}}=1.2$ while the
polymer is translocating it has enough time to fold itself. Therefore, almost
at all times even at the final time step of the translocation process
[panel(f)] the density is uniform. This is the feature of irregular
configurations of the polymer subchain inside the cavity at all times. This
leads to the wide distribution of the turning number for $R=10$ (black
circles) in Fig. 8(a). In contrast, for strong force $F_{\textrm{sp}}=15$ the
polymer dynamics is fast and it touches the front side of the cavity wall at
short time and the ring-shaped at the vicinity of the cavity wall is
constructed and as time passes this structure becomes thicker [red color
region in panel (l)]. Formation of the ring-shaped structure for the monomer
density in panel (l) is completely compatible with the presented result in
Fig. 8(a) for $R=10$ (orange diamonds), in which the probability distribution
of the turning number has two peaks at $\psi\approx 5$ and -5 indicating that
the active polymer with strong SP force has a regular structure at the end of
the translocation process. This behavior refers to the fact that, formation of
the spiral structure at the strong SP force regime $F_{\textrm{sp}}=15$ inside
a circular cavity with radius $R=10$, mostly, is from outer layer to the inner
one. On the other hand, in the weak SP force regime $F_{\textrm{sp}}=1.2$
during the folding, the polymer inside the cavity has enough time to explore
different spatial configurational states due to the thermal fluctuations,
therefore, the polymer is irregularly packed.

Figures 10 and 11 are the same as Fig. 9 but for different values of cavity
radius $R=20$ and 50, respectively. Comparing Figs. 10 and 11 with Fig. 9
reveals that by increasing the value of $R$ those effects discussed above for
small cavity radius $R=10$ are become less pronounced.

At weak force regime of $F_{\textrm{sp}}=1.2$, as the value of $R$ increases
the available space for the polymer sub-chain inside the cavity increases too.
This leads to funnel shape for the monomer density at short time limit [see
panel (b) in Fig. 10]. With the passage of the time the size of the funnel
grows and at the end of translocation process [panel (f) in Fig. 10] changes
to a pear shape with the larger density in the vicinity of the nano-pore and
smaller density close to the cavity wall specially next to cavity wall in
front of the nano-pore.

In the limit of strong SP force $F_{\textrm{sp}}=15$ with $R=20$ at short time
limit, e.g. $t=0.2\langle\tau\rangle$, as depicted in panel (h) of Fig. 10
similar to panel (b) a funnel shape is made. But in contrast to panels (c) to
(f), two separate regions with higher values of the monomer density evolve as
seen in panels (i) to (l). The first high density region is in the vicinity of
the nano-pore in which due to the high value of SP force the monomers are
accumulated in. The other high density region is the vicinity of the cavity
wall wherein the polymer sub-chain interacts with and spend more time there.
The results for the distribution of turning number (blue squares) in panels
(d) and (f) in Fig. 8 confirm that the configuration of the polymer inside
the cavity with radius $R=20$ at the end of translocation process is more
regular for the strong force $F_{\textrm{sp}}=15$ than that of the weak force
$F_{\textrm{sp}}=1.2$.

The results in Fig. 11 are similar to those of Fig. 10 but are less
pronounced due to the fact that there is more available space for $R=50$ as
mentioned above. Moreover, as seen in Fig. 11 the large value of the
available space leads to the irregular configurations at both weak and strong
force limits due to the lack of the spiral configurations, which is in
agreement with the results for the distribution of turning number (violet
diamonds) in panels (d) and (f) in Fig. 8.

##  IV Summary and conclusions

In summary we have investigated the dynamics of an active semi-flexible
polymer translocation into a circular cavity with the radius of $R$ in two
dimension using the Langevin dynamics simulation method. An active polymer has
been modeled by applying the SP force $F_{\textrm{sp}}$ on the specified
segments of the polymer. In addition to the force exponent the detail dynamics
of the process has been explored by the study of different quantities such as
waiting time distribution, velocities of monomers, turning number and monomer
density.

The most important quantity that reveals the global dynamics of translocation
process is the translocation time $\tau$. The relation between $\tau$ and the
SP force $F_{\textrm{sp}}$ introduces the force exponent $\beta$ as
$\tau\propto F_{\textrm{sp}}^{\beta}$. Our results show that the force
exponent $\beta$ has a non-monotonic behavior with respect to the cavity
radius. For the smallest value of $R=10$ in the present study, the value of
the force exponent is $\beta\approx-1$ and by increasing the value of $R$ the
value of $\beta$ grows and it gets its maximum at $R\approx R_{\textrm{g}}$,
where $R_{\textrm{g}}\approx 23$ is the average value of the radius of
gyration for a passive semi-flexible polymer in the free space in two
dimension. For large values of $R$, the $\beta$ approaches its asymptotic
value $\beta\approx-0.92$.

Figure 12: (a) The average radius of gyration $R_{\textrm{g}}$ as a function
of the normalized time $t/\langle\tau\rangle$ for fixed value of the cavity
radius $R=10$ and various values of the SP force $F_{\textrm{sp}}=15$ (orange
triangles left), 10 (blue triangles up), 5 (green diamonds), 2 (red squares)
and 1.2 (black circles). Panels (b), (c), (d) and (e) are the same as panel
(a) but for different values of $R=20$, 30, 40 and 50, respectively. (f) The
$R_{\textrm{g}}$ as a function of $t/\langle\tau\rangle$ for fixed value of
$F_{\textrm{sp}}=1.2$ and various values of $R=10$ (black circles), 12 (red
squares), 15 (green diamonds), 20 (blue triangles up), 30 (orange triangles
left), 40 (cyan triangles down) and 50 (violet triangles right). Panels (g),
(h), (i) and (j) are the same as panel (f) but for different values of
$F_{\textrm{sp}}=2$, 5, 10 and 15, respectively.

The waiting time distribution, which is the time that each monomer spends at
the nano-pore during the translocation process, illustrates the dynamics of
the process at the monomer level. The results of the present study show that
at constant value of the cavity radius, the waiting time decreases by
increasing the value of the SP force in both the weak as well as the strong
force limits [panels (a)–(d) in Fig. 4]. On the other hand, at constant value
of SP force and at weak force limit the waiting time is sensitive to the value
of $R$ [see panel (e) in Fig. 4], while at the strong SP force limit [panel
(h) in Fig. 4] the value of $w$ does not depend on the value of $R$ as the
main contribution to the dynamics of the translocation process comes from the
SP force.

The next quantity that has been considered is the $x$ component of the monomer
velocities as a function of the $s$ at different times during the
translocation process. At constant values of $R$ and $F_{\textrm{sp}}$ as time
passes the oscillations in the velocity curves appear [different rows, e.g.
panels (a)-(e), in Fig. 5]. The oscillations in the velocity curves become
more visible as the value of the SP force increases [e.g. panels (a) to (e) in
Fig. 5]. As the value of the cavity radius increases the wavelength of the
oscillations increases and the number of periods of the oscillations decreases
[see e.g. panels (c)-(h)-(m) or panels (e)-(j)-(o) in Fig. 5].

The above results indicate that as the value of $R$ decreases, the
configuration of the polymer inside the cavity become in average more regular.
To quantify this, the turning number $\psi$, which is the average of the
number of turns in the polymer configuration [see Eq. 5] has been studied. As
seen in Fig. 7(c) for small values of $R$ and strong force the high value of
$\langle|\psi|\rangle$ confirms that the polymer configuration is more regular
and makes the spiral form. In contrast, at large value of $R$ and small value
of the SP force, the polymer mostly has an irregular configuration.

To see the dynamics of the average spatial configuration of the polymer during
the translocation process the monomer number density has been investigated at
different times. For smallest value of cavity radius $R=10$ and the weak SP
force $F_{\textrm{sp}}=1.2$ the monomers are uniformly distributed inside the
cavity almost at all moments during the translocation process [except at the
beginning of the process, panel (b)–(f) in Fig. 9]. At the strong SP force
limit of $F_{\textrm{sp}}=15$ a ring structure is constructed and with the
passage of the time the thickness of the ring increases. This confirms that
the polymer configuration in average makes a regular spiral [panels (h)–(l) in
Fig. 9]. With the increase of cavity radius to $R=20$ and at the weak force
limit of $F_{\textrm{sp}}=1.2$, the density makes a pear shape and as time
passes the pear occupies the whole cavity space [panels (b)–(f) in Fig. 10].
At constant $R=20$ for the strong force $F_{\textrm{sp}}=15$ the density is
more pronounced in two regions in the cavity. The first region is the vicinity
of the nano-pore and the second one is next to the cavity wall in front of the
nano-pore [e.g. panel (i) in Fig. 10]. With the passage of the time both
regions extend [panels (h)–(l) in Fig. 10]. Especially at the end of the
translocation process the second region extend around the whole cavity wall
[panel (l) in Fig. 10]. With the increase in the cavity radius even further
to $R=50$, the less pronounced results are obtained compared to those for
$R=20$.

The results of the present study shed light on the deeper understanding about
the dynamics of the injection of an active polymers into a confined space such
as biological cells.

###### Acknowledgements.

A.R.-D. is indebted to Dr. Farshid Mohammad-Rafiee for his compassionate
supports.

##  Appendix A Radius of gyration

The time evolution of the radius of gyration (RG) is presented in this
appendix. The RG for a polymer configuration is obtained by calculating the
average distances of the monomers from the center of mass of the polymer.
Then, averaging over different trajectories gives a general information about
the average spatial size of the polymer. The explicit relation to calculate
the average of the RG is given by RubinsteinBook

$R^{2}_{\textrm{g}}=\frac{1}{N}\sum_{i=1}^{N}\langle(\vec{r}_{i}-\vec{r}_{\textrm{CM}})^{2}\rangle,$
(6)

where, $R_{\textrm{g}}$ is the average value of the RG, $\vec{r}_{i}$ and
$\vec{r}_{\textrm{CM}}=1/N\sum_{i=1}^{N}\vec{r}_{i}$ are position vector of
the ith monomer and center of mass vector of the polymer, respectively, with i
as the index of the monomers and $N$ is the total number of monomers in the
polymer. Here, $\langle\ldots\rangle$ denotes the ensemble average over
different trajectories. To unveil the time evolution of RG, in Fig. 12(a)
$R_{\textrm{g}}$ of the trans-side subchain has been plotted as a function of
the normalized time $t/\langle\tau\rangle$ for fixed value of the cavity
radius $R=10$ and for various values of the SP forces $F_{\textrm{sp}}=1.2$
(black circles), 2 (red squares), 5 (green diamonds), 10 (blue triangles up)
and 15 (orange triangles left). Panels (b), (c), (d) and (e) are the same as
panel (a) but for different values of $R=20$, 30, 40 and 50, respectively. In
panel (a), the curves for different values of $F_{\textrm{sp}}$ are almost
coincide each other, due to the small value of $R=10$ that leads to the small
available spatial space to the polymer. As the value of the cavity radius
increases, from panel (a) to (e), more spatial space is available to the
trans-side polymer subchain. Therefore, as the value of the SP force increases
the trans-side polymer subchain explores more space that yields the larger
value of the RG at longer time scales. Panel (f) demonstrates $R_{\textrm{g}}$
as a function of $t/\langle\tau\rangle$ for fixed value of SP force
$F_{\textrm{sp}}=1.2$ and various values of $R=10$ (black circles), 20 (blue
triangles up), 30 (orange triangles left), 40 (cyan triangles down) and 50
(violet triangles right). As can be seen by increasing the value of
$F_{\textrm{sp}}$, from panel (f) to panel (j), the curves deviate more from
each other. It means that for strong SP force [panel (j)] with increasing the
value of $R$, monomers explore more space, than that of the weak SP force in
panel (f).

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