# Controlling polymer translocation with crowded medium and polymer length
asymmetry

Vrinda Garg1 Rejoy Mathew1 Riyan Ibrahim1 Kulveer Singh1 Surya K. Ghosh1
<EMAIL_ADDRESS>1Department of Physics, National Institute of Technology,
Warangal, 506004, India

###### Abstract

Polymer translocation in crowded environments is a ubiquitous phenomenon in
biological systems. We studied polymer translocation through a pore in free,
one-sided (asymmetric), and two-sided (symmetric) crowded environments.
Extensive Langevin dynamics simulation is employed to model the dynamics of
the flexible polymer and crowding particles. We studied how crowding size and
packing fraction play a crucial role in the translocation process. After
determining the standard scaling properties of the translocation probability,
time, and MSD, we observed that the translocation rate and bead velocities are
location-dependent as we move along the polymer backbone, even in a crowd-free
environment. Counter-intuitively, translocation rate and bead velocities
showed the exact opposite behavior; for example, middle monomers near the pore
exhibit maximum bead velocity and minimum translocation rate. Free energy
calculation for asymmetrically placed polymer indicates there exists a
critical number of segments that make the polymer to prefer the receiver side
for translocation. For one-sided crowding, we have identified a critical
crowding size above which there exists a non-zero probability to translocate
into the crowding side instead of the free side. Moreover, we have observed
that shifting the polymer towards the crowded side compensates for one-sided
crowding, yielding an equal probability akin to a crowder-free system. Under
two-sided crowding, the mechanism of how a slight variation in crowder size
and packing fraction can force a polymer to switch its translocation direction
is proposed, which has not been explored before. Using this control we
achieved an equal translocation probability like a crowd-free scenario. These
conspicuous yet counter-intuitive phenomena are rationalized by simple
theoretical arguments based on osmotic pressure and radial entropic forces.

##  I Introduction

The transport of a polymer through a pore is omnipresent in many biological
systems. Examples include RNA passing through a pore created by a membrane-
bound protein Butler  _et al._ (2006); Dubbeldam  _et al._ (2007), RNA and DNA
sequencing Meller  _et al._ (2000); Howorka  _et al._ (2001); Vercoutere  _et
al._ (2001); Singh  _et al._ (2013); Nakane  _et al._ (2004); Muthukumar
(2001), polymer transport processes Zhou and Wang (2022); Citovsky and
Zambryski (1993); Singh and Sain (2013); Theeyancheri  _et al._ (2022), gene
therapy Di Marzio and Mandell (1997); Doi  _et al._ (1988); Sung and Park
(1996), viral ejection Kasianowicz  _et al._ (2002); Peskin  _et al._ (1993);
Kasianowicz  _et al._ (1996); Pavan K Madasu and Chandran (2023), controlled
delivery Liechty  _et al._ (2010); Bettini  _et al._ (2001); Roy  _et al._
(2023), and polymer sorting and ultrafiltration Ge and Wu (2010). In in-vitro
setups, translocations are mainly induced by driving forces, such as external
applied electric field, Ali and Yeomans (2005); Kasianowicz  _et al._ (1996);
Lubensky and Nelson (1999a); Sarabadani  _et al._ (2022); Hsiao (2020);
Buyukdagli  _et al._ (2019); Lathrop  _et al._ (2010), controlling the
translocation of a single molecule through a glass nanopore on a 3D
nanopositioner Leitao  _et al._ (2023); Sahoo and Chakrabarti (2023), pulling
force exerted on polymer’s end Kantor and Kardar (2004); Ollila  _et al._
(2009); Panja and Barkema (2008), binding particles (chaperones) Ambjörnsson
and Metzler (2004); Luo  _et al._ (2006); Yu (2020); Yu and Luo (2011), are
known as forced translocation Lubensky and Nelson (1999b). In many biological
systems, unforced translocations occur naturally dictated by conformational
entropy Kasianowicz  _et al._ (1996, 2002); Lehtola  _et al._ (2010); Simon
_et al._ (1992). Despite its omnipresence and importance in biological
processes, unforced translocation has received less attention Lehtola  _et
al._ (2010); Simon  _et al._ (1992). Especially the system’s intrinsic
parameters, such as the symmetry of the length distribution of the polymer
being translocated and the effects of crowding present in the cellular
environment. Here, we investigate the generic behavior of self-avoiding
flexible polymersGhosh  _et al._ (2009) exhibiting unforced translocation
through a pore for free and crowded environments. A deeper understanding of
these natural unforced processes can help in designing more efficient
practical in-vitro setups.

Polymer translocation through pores displays a broad range of scaling regimes,
as shown by numerical simulations Luo and Metzler (2010a); Wong and Muthukumar
(2008); Lucas  _et al._ (2022); Luo  _et al._ (2008), analytical theories
Gopinathan and Kim (2007); Wen and Zhang (2021), and experiments Kasianowicz
_et al._ (1996). The average translocation time $\tau$ as a function of the
chain length $N$ is an important measure of the underlying dynamics.
Especially in the case of unforced translocation, the barrier is so high that
it is almost practically impossible to have a successful translocation solely
due to the thermal agitation for long polymer chains in the limit of $N\gg 1$
Kantor and Kardar (2004). As the repeated chemical units of the polymer pass
through the pore, they encounter a depletion in their available conformational
entropy creating an overall entropic barrier resulting in constrained
diffusion. According to Kramers’s analysis of diffusion across an entropic
barrier, the translocation time is scaled as $\tau\sim N^{\alpha}$, for
phantom chains in the case of unforced translocation the scaling exponent is
$\alpha=2$ ($\tau\sim N^{2}$) and for the forced translocation $\alpha=1$
($\tau\sim N$) Kramers (1940). The Rouse model for the dynamics of phantom
chains also predicts a time of the order of $N^{2}$ for equilibration. For a
phantom chain, translocation time depends on the relative magnitudes of three-
time scales in terms of dimensionless factor that characterizes the
translocation process $\tilde{\tau}$ and medium’s viscosity $\eta$: total
translocation time $\tau\sim(b^{2}/D_{p})\tilde{\tau}$, polymer relaxation
time $\tau_{R}\sim(\eta b^{3}/k_{B}T)N^{2}$, and
$\tau_{0}\sim\phi^{-2/3}/D_{0}$ for obstacles motion, where Gaussian polymer
(phantom chain) of length $N$ (in units of the Kuhn length $b$) has diffusion
coefficient $D_{p}$, $D_{0}$ is the diffusion coefficient of obstacles, and
$\phi$ is the volume fraction of randomly distributed obstacles. Two regimes
exist depending on the relative mobility of the obstacles: the dynamic
obstacles regime where $\tau_{0}\ll\tau_{R}\ll\tau$ and the static obstacles
regime where $\tau_{R}\ll\tau\ll\tau_{0}$ Sung and Park (1996); Doi  _et al._
(1988); Gopinathan and Kim (2007); Chen and Luo (2013); Kantor and Kardar
(2004); Lehtola  _et al._ (2010); Luo and Metzler (2010b). With excluded
volume, in the presence of an external driving force $f$, the translocation
time $\tau$ scales as $\tau\sim N^{\beta}/f^{\gamma}$ where $\gamma$ is the
scaling exponent associated with the external forces. Biased translocation
exhibits anomalous behavior with various values of both these exponents in
several experimental Meller and Branton (2002); Molcrette  _et al._ (2022) and
computational studies Chuang  _et al._ (2001); Afrasiabian and Denniston
(2020). Once self-avoidance is included, translocation time increases
dramatically with a scale of $\alpha=2.5$ ($\tau\sim N^{2.5}$), which is an
exponent of Rouse relaxation of self-avoiding chain in 2D Kantor and Kardar
(2004); Chuang  _et al._ (2001). For the self-avoiding chain, the
translocation time is longer than the Rouse relaxation time
($\tau>>\tau_{R}$). According to the flory-exponent theory, the scaling of
translocation time for a self-avoiding chain is given as $\tau$ $\sim$
$N^{\beta}$ where $\beta$ = 1 + 2 $\nu$ and with the flory exponent $\nu=0.75$
($\tau\sim N^{2.5}$) in 2D and $\nu=0.588$ ($\tau\sim N^{2.176}$) in 3D Luo
_et al._ (2006); Luo and Metzler (2010b); Chen and Luo (2013). Likewise, the
scaling for MSD has been noted. It takes the power law form: $\langle\Delta
r^{2}(t)\rangle\sim t^{\alpha}$, where $\alpha$ is the anomalous diffusion
exponent. It is observed that $\alpha$ = 0.8 resembles the experimental value
Lehtola  _et al._ (2010); Chuang  _et al._ (2001); Chaudhury and Cherayil
(2008). The shapes and sizes of the fluidic channel and cavity Polson and
Heckbert (2019) also have a strong effect when a polymer is translocated into
or out of a confined environment Luo and Metzler (2010a); Luo  _et al._
(2009); Luo and Metzler (2010b); Sarabadani  _et al._ (2022).

In this work, we considered a self-avoiding flexible polymer in an unbiased
and unforced environment, threading its way through the pore as a fundamental
model and as a first step towards understanding the translocation dynamics in-
vivo set-ups. Here, we specifically focused on the system’s intrinsic
parameters rather than the external influences. We examined certain parameters
to elucidate translocation dynamics in this system thoroughly. Quantitative
measurements of the probability of translocation events and the corresponding
translocation times through the pore provide a fundamental basis for our study
and validate the process. To understand the diffusion mechanism, we
investigate the MSD of the center of mass of polymer, head, middle, and tail
monomers. A study on how different segments of the polymer are moving across
the pore is drawn to attention. We examine the quantities, such as the average
bead crossing time of individual monomers, the translocation rate of each
bead, and their translocation speed across the pore. In particular, we find a
counterintuitive trend of monomers’ translocation rate and translocation speed
as they pass through the pore.

As the polymer translocates through the pore, the number of accessible
conformations significantly reduces. This leads to a decrease in the chain’s
conformational entropy and an increase in its free energy. We examined the
free energy profile for the no-crowding case and observed a symmetric free
energy barrier when the polymer chain is placed symmetrically at the pore.
This analysis resembles the analytical calculations Muthukumar (2016).
Further, we find that an asymmetrically placed polymer leads to a critical
number of segments that, when translocated, causes the free energy minima and
polymer to prefer the receiver side.

The highly crowded environment of actual biological cells containing large
macromolecules like proteins, lipids, ribosomes, and cytoskeleton fibers
features volume occupancies up to $\phi\sim 40\%$. Studying the impact of
crowding is crucial for comprehending translocation in realistic cellular
environments due to high cell densityMuthukumar (2016); Gopinathan and Kim
(2007); Polson and Heckbert (2019); Shin  _et al._ (2015). A number of recent
works have shed light on the translocation processes in the presence of non-
inert crowder (chaperons)Yu and Luo (2011); Ambjörnsson and Metzler (2004). An
aspect that has received very little attention is the impact of inert
crowding, resulting in a biased but still unforced system by introducing
crowders at one side of the box, leading to asymmetric crowding and on both
sides of the box, creating a symmetric crowded environment as a second step to
understand the translocation dynamics in crowded in-vivo set-ups. Polymer
threading its way through such an environment is subjected to an entropic
penalty, and interaction between crowder-polymer affects the translocation
dynamics. Here, we quantify the effects of modifying the length distribution
of polymer from its initial configuration, varying the size and packing
fraction of the crowders, and show that it significantly affects the
translocation dynamics. We find that when the crowders size reaches a
crossover and deviates from the size of each chemical unit of the polymer, the
polymer-crowder interaction fundamentally changes the direction of
translocation and shows a sudden jump within two extremes of probability.
Moreover, we show that by tuning the crowded environment, we can control the
dynamics and switch the direction of the translocation process, which can be
applied for better drug delivery, DNA sequencing, and transport processes.

The article is constructed as follows: Section II elucidates the methods and
models employed in this study. In section III, we present the simulation
results and discussions for the three distinct modeling frameworks. Firstly
(section III A), we examine the dynamics of polymer translocation without
crowder. Next (section III B), we investigate the impact of the crowder on one
side of the box. Finally (section III C), we study the translocation process
driven by the crowders on both sides of the box. Section IV summarizes the
conclusions drawn from this work.

Figure 1: Schematic illustration of the system of polymer translocation
process through a pore. The wall is purely repulsive and consists of immobile
LJ particles, and has a width $\sigma$ the same as the size of each repeated
unit of polymer (monomer). As an initial configuration, the polymer is placed
symmetrically at the pore center. The left part represents the cis-
translocation of polymer, and the right part represents the trans-
translocation of polymer of length, $N=65$. Here, we show the initial
configuration of all the systems with a polymer being symmetrically placed at
the pore. (a) Polymer Translocation in a crowd-free environment. (b) Polymer
translocation in the presence of one-sided crowding. Yellow particles on the
trans side represent crowders. (c) Polymer translocation when crowders are
present on both sides of the box. Packing fractions of crowders on both sides
is kept the same. We are changing the size of the crowders on the trans side,
which is depicted in yellow, while the crowders on the cis side are shown in
blue.  Figure 2: (a) Translocation probability P of a free polymer of length
$N=21,~{}43,~{}65,$ and $131$. $P_{c}$ represents the translocation
probability to the cis-side (circle) while $P_{t}$ shows the translocation
probability to the trans side (triangle). (b) Scaling of the translocation
time $\tau$ with respect to the polymer of length $N=21,~{}43,~{}65,$ and
$131$ for no crowding case. Average $\tau$ shows the same behavior in the case
of cis and trans translocation for varying lengths of polymer with the scaling
of $\tau_{c}\sim\tau_{t}\sim N^{2.5}$. (c) The mean-squared displacement of
the first monomer $g_{head}(t)$, middle monomer $g_{mid}(t)$, end monomer
$g_{tail}(t)$ and center of mass $g_{3}(t)$ of the polymer of $N=65$. The
curve shows alike behavior irrespective of the side (cis/trans), giving
$\alpha=0.8$ in 2D at long times.

##  II Methods and Modeling

We studied the unforced translocation process of a flexible self-avoiding
polymer in a free and crowded two-dimensional (2D) environment. The polymer of
length $L$ is represented by a space curve $\textbf{r}(s)$ in 2D, where the
parameter $s$ goes from $s=0$ to $L$, along the chemical distance or contour
length of the chain. The standard two-dimensional coarse-grained bead-spring
model of the polymer chain is used to represent the translocating polymer. In
the discrete limit, the polymer consists of $N+1$ monomers of size $\sigma$
which are connected by $N$ Finite Extension Nonlinear Elastic (FENE) springs,

$U_{FENE}(l_{ij})=\begin{cases}-\frac{1}{2}kl_{0}^{2}\ln\left(1-\frac{l_{ij}^{2}}{l_{0}^{2}}\right),&\text{for~{}}l_{ij}\leq
l_{0}\\\ \infty,&\text{otherwise}.\end{cases}$ (1)

Here $k$ is the FENE spring constant, $l_{ij}$ is the instantaneous
separation, and $l_{0}$ is the maximum allowed length between two consecutive
monomers. In the discrete picture, the $i^{th}$ bead is represented by a 2D
position vector $\textbf{r}_{i}(t)$. The excluded volume interaction
exclusively between the beads, crowders, and bead-crowders is modeled by a
standard truncated short-range repulsive Lennard-Jones (LJ) potential,
commonly known as Weeks-Chandler-Andersen (WCA) potential as Weeks  _et al._
(1971)

$U_{LJ}({r_{ij}})=\begin{cases}4\epsilon\left[\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6}\right]+\epsilon,\quad&\text{for
~{}}r_{ij}\leq 2^{1/6}\sigma_{ij}\\\ 0,\quad&\text{otherwise}.\end{cases}$ (2)

Here $r_{ij}=|\textbf{r}_{i}-\textbf{r}_{j}|$ is the distance between $i$ and
$j$ particle, $\sigma_{ij}=(\sigma_{i}+\sigma_{j})/2$ is the diameter of the
effective interaction region of the interacting pairs with diameter
$\sigma_{i},\sigma_{j}$ and $\epsilon$ is interaction strength of the
potential. The use of a purely repulsive part of the LJ potential mimics the
good solvent by cutting out the attractive effect.

We consider a two-dimensional rectangular box separated by a repulsive wall in
the middle consisting of rigid repulsive LJ particles of thickness the same as
the diameter of a monomer with bead size $\sigma=1$. The middle wall separates
the box into two equal regions; the left side is referred to as cis while the
right side is called trans. These two separate regions cis and trans are
connected by a pore of length $w_{H}=\sigma$ and pore width $w_{V}=1.6\sigma$
at the center of the wall Chen and Luo (2013). A study of the effect of box
size has been done. The box has been chosen to be big enough so the polymer
does not feel any effect from the box size. We have checked the effect of box
size for different scaling parameters, such as translocation time and
probability, and our model matches with both experimental and analytical
theories of scaling regime. The geometry of the cavity plays a crucial role
and has been studied extensively Polson and Heckbert (2019); Sharma  _et al._
(2022); Saltzman and Muthukumar (2009); Palyulin  _et al._ (2014); Ikonen  _et
al._ (2012); Slonkina and Kolomeisky (2003).

In our simulation, the equation of motion of the dynamics of the $i^{th}$
monomer is governed by the Langevin equation by neglecting hydrodynamic
interactions among the monomers as

$\begin{split}m\frac{d^{2}\textbf{r}_{i}(t)}{dt^{2}}&=-\xi\textbf{v}_{i}(t)+{\textbf{F}_{i}^{R}}(t)-\sum_{j=1,~{}i\neq
j}^{N+1}\bm{\nabla}U_{LJ}(|\textbf{r}_{i}-\textbf{r}_{j}|)\\\
&-\bm{\nabla}U_{FENE}(|\textbf{r}_{i}-\textbf{r}_{i\pm
1}|)-\sum_{j=1}^{N_{c}}\bm{\nabla}U_{LJ}(|\textbf{r}_{i}-\textbf{r}_{j}^{c}|)\\\
&-\bm{\nabla}U_{LJ}(|\textbf{r}_{i}-\textbf{R}_{Wall}|)-\bm{\nabla}U_{LJ}(|\textbf{r}_{i}-\textbf{R}_{Box}|),\end{split}$
(3)

where $\textbf{r}_{i}$ is the position, $m$ is the mass, $\textbf{v}_{i}$ is
the velocity of the $i^{th}$ monomer, $N_{c}$ is the number of crowder,
$\textbf{r}_{j}^{c}$ is the position of the $j^{th}$ crowding particle and
$\xi$ is the friction coefficient, and thermal fluctuation is considered by
$F_{i}^{R}$ which is the random forces satisfying fluctuation-dissipation
theorem,

$\langle{\textbf{F}}_{i}^{R}(t)\rangle=0,\langle{\textbf{F}}_{i}^{R}(t)\cdot{\textbf{F}}_{j}^{R}(t^{\prime})\rangle=2dk_{B}T\xi\delta_{ij}\delta(t-t^{\prime}),$
(4)

that connect the particle diffusivity to the friction coefficient
$D=k_{B}T/\xi$. Here $d$ is the dimension of the system, in our case $d=2$,
$k_{B}$ is the Boltzmann constant, $T$ is the temperature, and $i$ and $j$
represent the coordinate components. The presence of the confining rectangular
box is represented by $\textbf{R}_{Box}(x,y)$, where $x\in[-L_{x},L_{x}]$,
$y\in[-L_{y},L_{y}]$ and $L_{x}=2L_{y}$. The wall in the middle is represented
by $\textbf{R}_{Wall}(x=0,y)$ and the pore is represented by
$\textbf{R}_{pore}(x=0,y=0)$.

Similarly, the corresponding Langevin equation Luo  _et al._ (2006) for the
dynamics of the $i^{th}$ crowding particle can be written, where $m_{i}^{c}$
is the $i^{th}$ crowding particle’s mass, $r_{i}^{c}(t)$ is crowder position
at time $t$, and other crowding particles at position $r_{j}^{c}(t)$,
$\xi_{i}$ is friction coefficient, $v_{i}^{c}$ is the $i^{th}$ crowder’s
velocity. The sizes of a crowder are represented as $\sigma_{c}$ for the cis-
side and $\sigma_{t}$ for the trans-side. The friction and mass of the crowder
on each side are proportional to their sizes, with
$\frac{\xi_{t}}{\sigma_{t}}=\frac{\xi_{c}}{\sigma_{c}}$ and
$\frac{m_{t}}{\sigma_{t}^{2}}=\frac{m_{c}}{\sigma_{c}^{2}}$ respectively.

We are using LJ parameters m, $\sigma$, $\epsilon$ for defining mass,
distance, and energy scales corresponding to $t=\sqrt{m\sigma^{2}/\epsilon}$
and $f=\epsilon/\sigma$ for time scales and force scales in $ps$ and $pN$
respectively. The dimensionless parameters used in the simulation are
temperature $T=1.2$, the maximum allowed separation between each monomer
$R_{0}=1.5$, spring constant $k=30$, friction $\xi=0.7$, the mass of each
monomer $m=1$.

For each translocation process, as a starting configuration, we placed the
middle monomer $(N/2+1)$ at the center of the pore. We keep the middle monomer
fixed until the rest of the polymer equilibrates for $10^{5}$ time steps. All
the simulations are performed using a Langevin thermostat, to begin with, a
truly random initial configuration. The equation of motion is integrated using
the velocity Verlet algorithm in each step. After the equilibration time, the
middle monomer is released to allow the translocation process. After each
successful translocation event, where the polymer ends up moving completely to
either side of the wall, the simulation is stopped. In order to get
satisfactory statistics, we averaged our data over $10^{3}$ independent
realizations, with a time step $\Delta t=0.005$.

Figure 3: (a) For a polymer chain of $N$ monomers labeled
$i=1,~{}2,~{}3,\cdots,~{}N$ having center at $N/2+1$, its left-side can be
represented as $j_{L}=i,1\leq i<(N/2+1)$ and right-side of the chain as
$j_{R}=(N/2+1)-i$, $(N/2+1)\leq i\leq N$. Monomer id w.r.t. center bead
$N/2+1$ of the polymer against average bead crossing time $\tau_{crossing}$ of
polymer length $N=65$ (blue), $~{}131$ (orange) is plotted. (b) Translocation
rate $k_{T}$ of a polymer of length $N=65,~{}131$. The graph is plotted
symmetrically w.r.t to the center bead at the pore. Curves on the left
represent the translocation rate of $j_{L}$ to show trans-translocation
(triangles), and the right curve represents the translocation rate of $j_{R}$
to show cis-translocation (circle). Figure 4: Bead velocity $v_{b}$ of the
individual monomer for a polymer of length $N=65,~{}131$. The graph is plotted
symmetrically w.r.t. center bead ($N/2+1$) at the pore and indicates bead
velocity when moving to the trans-side (triangle) and cis-side (circle).
Figure 5: Plot of probability of translocation vs. the number of monomers
being shifted to the trans side. Here, $N_{trans}$ is the length of the
polymer on the trans side, and $(N-1)/2$ represents the middle of the polymer
length $N$. The probability of overcoming the nucleation barrier for further
successful translocation for asymmetrically placed polymer towards the trans-
side is plotted. Asymmetry introduced in the polymer, which was initially
placed symmetrically at the pore, is represented in terms of the number of
monomers shifted from its middle to the trans side and shown as
$N_{trans}-(N-1)/2$. For symmetrically placed polymer, $N_{trans}-(N-1)/2=0$,
which gives $N_{trans}=(N-1)/2$. The probability of cis-side (circle) and
trans-side (triangle) for the different $N$ is shown. It starts from $P=0.5$
for the symmetrically placed polymer when $N_{trans}=(N-1)/2$ and reaches
saturation when a sufficient amount of monomers has been shifted for the
nucleation phenomenon. Figure 6: Plot of Free energy $F(n)$ vs. $n/N$, where n
represents the number of asymmetrically placed segments, and N is the polymer
length. Here $n$ varies from the initial monomer of the polymer chain to its
entire length. $n^{*}$ is the critical number of polymer segments representing
the free energy maxima. Figure 7: Here, $n^{*}$ is the critical number of
polymer segments which need to be translocated for successful translocation
event and represent the free energy maxima, $N$ is polymer length and
$\Delta\mu$ is the chemical potential gradient between cis and trans-side. The
black dashed line represents no crowding case where $\Delta\mu=0$ and
$n^{*}/N$ is exactly half for all chain lengths $N$, and the free energy
barrier is symmetric. The chemical potential gradient varies from negative to
positive, shown by blue to red.  Figure 8:  Change in n∗ against $N\Delta\mu$.
Plot showing the dependence of the critical number of polymer segments
translocated on the trans side $n^{*}$ on $N\Delta\mu/k_{B}T$ for the polymer
of different lengths $N=43,~{}65,~{}91,~{}101$. When $\Delta\mu=0$, which
leads to n${}^{*}=N/2$ as we move towards the positive side, n∗ decreases, and
all the N collapses. This graph considers two sides of the wall: the receiver
and the donor. The chemical gradient is calculated from the difference between
the donor and receiving side, and the sign decides the direction in which
translocation could be preferred.

##  III RESULTS AND DISCUSSION

###  III.1 Translocation without crowders

####  III.1.1 Translocation probability (P), time ($\tau$), and MSD

We first considered the polymer translocation in a crowd-free medium (Fig. 1a
). We calculated the translocation probability $P$ and time $\tau$. The
translocation probability $P$ can be defined as the ratio of successful
translocation events towards a particular side to the total number of
successful events Lehtola  _et al._ (2010); Luo  _et al._ (2006). The
probability of the polymer to translocate to cis (left) side is represented by
$P_{c}$ and to the trans (right) side is $P_{t}$. The time required for a
polymer positioned symmetrically with a center bead initially at $t=0$ in the
middle of the pore to fully translocate to either side of the wall is referred
to as translocation time $\tau$. If the polymer goes to the cis (left) side,
we call it a cis-translocation time $\tau_{c}$, and for trans (right) side, it
is a trans-translocation time $\tau_{t}$.

Ideally, in the absence of crowders (Fig. 1a), the system remains unbiased
from either side and exhibits an equal probability of translocation Lehtola
_et al._ (2010). As expected, from our model, we have observed equal cis and
trans translocation probability, $P_{c}\approx P_{t}\approx 0.5$, for
different polymer length $N$ (Fig. 2a). The translocation time $\tau$ scales
as $\tau_{c}\sim\tau_{t}\sim N^{2.5}$ (Fig. 2b), as we vary length $N$,
exactly matches with analytical and previously found simulation studies on
unforced free polymer translocation Gopinathan and Kim (2007); Chuang  _et
al._ (2001).

To understand the dynamic behavior of the polymer, we analyzed the time
evolution of mean squared displacement (MSD) of the center of mass $g_{3}(t)$
and individual monomers: the first monomer $g_{head}(t)$ for $s=1$, mid
monomer $g_{mid}(t)$ for $s=N/2+1$, and end monomer $g_{tail}(t)$ for $s=N+1$.
In general, the MSD takes the following power law form: $\langle\Delta
r^{2}(t)\rangle\sim t^{\alpha}$, where $\alpha$ is the anomalous diffusion
exponent. Each danging end outside the pore has a characteristic relaxation
time $\tau_{R}$ much faster than that of the monomer moving through the pore.
The relaxation time (Rouse time) $\tau_{R}$ of a polymer is typically defined
as the characteristic time it takes to diffuse a distance of the order of its
size Rubinstein  _et al._ (2003). At small time scales when time $t$ is much
less than the relaxation time $\tau_{R}$ ($t<\tau_{R}$), the movement of the
individual monomers is different from the center of mass movement, resulting
in a significant difference in their $MSD(t)$. At a later time, when $t$ is
greater than relaxation time $\tau_{R}$ $(t>\tau_{R})$, the MSDs of the middle
monomer coincide with the center of mass. For the translocation of a self-
avoiding chain in 2D, the exponent can be written as $\alpha=2/(1+2\nu)$ where
$\nu$ is the flory swelling exponent. In the long time limit, the $g_{mid}(t)$
and $g_{3}(t)$ converge with $\alpha=0.8$ for $\nu=0.75$ (Fig. 2c) Chuang  _et
al._ (2001); Lehtola  _et al._ (2010). Moreover $g_{head}(t)$ and
$g_{tail}(t)$ also show similar behavior with exponent $\alpha=0.8$. However,
as the two ends of the polymer danging in bulk with more translational degree
of freedom, the MSD values are always greater than the other or center of mass
monomers.

####  III.1.2  Translocation Rate, Crossing Time and, Bead Velocity

During translocation processes, it is observed that different segments of the
polymer move differently. To envision this interesting phenomenon, we
calculated the average bead crossing time $\tau_{crossing}$, translocation
rate $k_{T}$, and bead velocity $v_{i}(t)$ for different segments of the
polymer. Our focus centers on investigating and highlighting the translocation
rate and bead velocity of an individual monomer, excluding considerations for
the polymer as a whole.

The average time taken by a monomer to translocate through the pore is defined
as the average bead crossing time $\tau_{crossing}$. Fig. 3a shows the
variation in crossing time taken by each monomer to pass the pore starting
from their respective initial position in bulk. As expected, for different
monomers, as we move along the backbone of the chain, the crossing times are
different. Due to the closeness of the pore, the central monomers take a
shorter time to pass the pore over their faraway peers, the tail monomers. In
spite of this trivial realization, if we look closely, we can observe that the
rate of change of $\langle\tau_{crossing}\rangle$ with respect to monomer id
is not constant but rather increases as it moves towards the tail. This
interesting feature will be addressed during the discussion of $k_{T}$ in the
following section.

The translocation rate $k_{T}$ is defined as the number of beads passing
through the pore per unit time. $k_{T}$ is calculated by taking the derivative
of the curve in Fig. 3a. As the translocation process starts with the center
bead $(N/2+1)$ at the pore, the translocation rate increases slowly for beads
near the center of the polymer chain due to alike entropic force from both
sides. The small values of $k_{T}$ in the middle clearly indicate that during
the translocation event, the monomers in the middle region have several back-
and-forth movements around the pore before they fully translocate to one side.
Subsequently, once the translocation of some significant middle portion of the
polymer has taken place, follow-up monomers motion is governed by the already
translocated part, leading to a higher translocation rate for the tail
monomers (Fig. 3b). The ratio of the translocation rates of the center monomer
with respect to the polymer end monomer are defined as
$\overline{k}_{L}={k_{T}(n_{first})}/{k_{T}(n_{center})}$ for the cis-side,
$\overline{k}_{R}={k_{T}(n_{end})}/{k_{T}(n_{center})}$ for the trans-side,
and the average translocation rate $\langle
k_{0}\rangle=(\overline{k}_{L}+\overline{k}_{R})/2$. The average translocation
rates, $\langle k_{0}\rangle$, for the initial and end monomers on the cis
($\overline{k}_{L}$) and trans ($\overline{k}_{R}$) sides relative to the
central monomer exhibit almost similar ratios of approximately 1.4 and 1.3 for
lengths $N=65,131$, respectively (see Table 1).

$~{}~{}~{}~{}~{}N~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}\overline{k}_{L}~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}\overline{k}_{R}~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}\langle k_{0}\rangle~{}~{}~{}~{}~{}$
---|---|---|---
65 | 1.37 | 1.49 | $\sim{1.4}$
131 | 1.32 | 1.25 | $\sim{1.3}$
Table 1: Motion of the center monomer with respect to the polymer end monomers
on the cis-side and trans-side is depicted as the ratio of translocation rate
and represented as $\overline{k}_{L}$ and $\overline{k}_{R}$ respectively. The
average translocation rate, $\langle k_{0}\rangle$, is the mean of
$\overline{k}_{L}$ and $\overline{k}_{R}$. We observe a similar average
translocation rate ratio for different polymer lengths $N$.

Now let us look at the individual bead velocity $v_{i}(t)=dr_{i}(t)/dt$, which
tracks the dynamics of the movement of the $i^{th}$ bead during a
translocation process. We have observed that middle monomers move faster than
tail monomers (Fig. 2c). Being close to the pore, the monomers in the central
region are restricted to exhibit quasi-one-dimensional diffusion from the
beginning of the translocation process until a successful crossover of that
bead happens. While for most of the translocation time, the tail monomers
undergo 2D diffusion until they approach the vicinity of the pore, resulting
in lower $v(t)$. From the MSD Study in Fig. 2c and bead velocity in Fig. 4c,
it can be observed that initially, the dangling tail monomers wander in the
medium for a reasonable amount of time and diffuses slowly. In contrast, the
middle monomers in the vicinity of the pore, part of quasi-one-dimensional
diffusion, diffuses faster, supporting the notion that resulted in higher
$v(t)$ for middle monomers than the tail monomers during the translocation
process Afrasiabian and Denniston (2020); Rezaie-Dereshgi  _et al._ (2022).
The monomer velocity dramatically increases when it approaches the vicinity of
the pore.

####  III.1.3 Asymmetrically placed polymer

Until now, in the free environment, we have studied translocation by always
placing the polymer symmetrically between the cis and trans side, with the
middle monomer at the pore as the initial configuration. Now, we are breaking
the symmetry by asymmetrically placing the polymer from its middle with extra
length into the trans side. As a result, we are designing the initial
configuration so that when the translocation process starts, the polymer
portion on the trans side is larger than the cis side. We want to know what
the free energy of the chain is as the polymer translocates from one region to
another.

We are interested in examining the impact of the asymmetrically placed polymer
on translocation probability see Fig. 5. When a polymer translocates through
the pore, the chain goes through a decrease in its conformational entropy, and
the excluded volume interactions become more pronounced, increasing the free
energy of the chain. Here, pore size is assumed to be short in comparison with
the length of polymer molecules that can be considered as a small hole in an
infinite wall and can allow only one monomer to pass through at a time Sung
and Park (1996); Muthukumar (1999); Di Marzio and Mandell (1997); Lehtola  _et
al._ (2010). The two translocating polymer segments on the cis and trans side
can be treated as two thermodynamic ensembles separated by the wall. At any
state, let there be $n$ segments of the translocating polymer in the trans-
side referred to as reciever and $N-n$ segments in the cis-side as donor, both
are in separate thermal equilibrium. The terms ”donor” and ”receiver”
designate the sides for polymer translocation, with the donor being where the
polymer originates and the receiver being the target (translocated) side. In
our study, we named the cis side as the donor and the trans side as the
receiver for clarity. In the context of observing free crowders, side
nomenclature can be chosen for convenience. The partition sum $Z$ consisting
of the total number of conformations of the chain can be written as
$Z=Z_{d}(N-n)~{}Z_{r}(n).$ The partition sum $Z(N)$ for a long tail $N>>1$ of
$N$ segments in the semi-infinite space (half) bounded by an impenetrable wall
to which one end is anchored is given as $Z_{half}(N)=\bar{z}^{N}N^{\gamma-1}$
where $\bar{z}$ is the effective coordination number for the orientation of
adjacent bonds and is commonly known as the connective constant. It can be
alternatively written as $\exp(-\mu/k_{B}T)$, where $\mu$ is the chemical
potential per segment. $\gamma$ is the critical exponent and depends on the
nature of the polymer and the solution. To mimic a good solution condition for
a self-avoiding polymeric chain, we have $\gamma$= 0.69. The Helmholtz free
energy is $F_{half}(N)=-k_{B}TlnZ_{half}(N),$ elaboratively can be written as
$\frac{F_{half}(N)}{k_{B}T}=\frac{\mu N}{k_{B}T}+(1-\gamma)\ln N.$ The
logarithmic part plays a significant role in establishing a free energy
profile for polymer translocation. The total free energy of the chain $F(n)$,
with $n$ segments translocated into the receiver-side can be written as the
sum of free energies of the two tails,

$\frac{F(n)}{k_{B}T}=(1-\gamma_{d})ln(N-n)+(1-\gamma_{r})ln(n)-\frac{n\Delta\mu}{k_{B}T},$
(5)

where $\gamma_{d}$ and $\gamma_{r}$ are critical exponents in donor and
receiver region respectively. $\Delta\mu=\mu_{d}-\mu_{r}$, where $\mu_{d}$ and
$\mu_{r}$ are the chemical potentials of the polymer segments in the donor and
receiver region respectively. The first two terms on the RHS emerged from the
entropy of two tails and clearly resulted in the free energy barrier. In
general, for this free energy barrier with its maximum value $F^{*}$, there
exists a critical number of translocated segments to receiver side, that is,
$n^{*}$ which can be obtained as the solution of $\partial F(n)/\partial n=0$,
gives

$\frac{n^{*}}{N}=\frac{(\tilde{\mu}+2-\gamma_{d}-\gamma_{r})}{2\tilde{\mu}}-\frac{\sqrt{(\tilde{\mu}+2-\gamma_{d}-\gamma_{r})^{2}-4\tilde{\mu}(1-\gamma_{r})}}{2\tilde{\mu}}$
(6)

where $\tilde{\mu}=N\Delta\mu/k_{B}T$. For the crowd-free case, the critical
value of $n^{*}$ can be obtained by putting $\Delta\mu=0$ in Equation 11. In
our model, we are taking $\gamma_{d}=\gamma_{r}=\gamma$. Then, Equation 9 can
be written as

$\frac{F(n)}{k_{B}T}=(1-\gamma)ln[n(N-n)].$ (7)

The critical number of translocated segments to receiver side $n^{*}$ for
$F^{*}$ is

$\frac{\partial F(n)}{\partial n}=\frac{(1-\gamma)(N-2n)}{n(N-n)},$ (8)

solution of $\partial F(n)/\partial n=0$ gives $n^{*}=N/2$. Fig. 6 shows
$F(n)$ vs. $n$, which represents the free energy landscape of polymer
translocation in the crowd-free case as a function of the translocation
extent. This can be interpreted as when there is a slight deviation from the
maxima of $F^{*}$ at $n^{*}=N/2$, the free energy landscape is downhill on
both sides (Fig. 6). The free energy barrier is symmetric when a polymer is
symmetrically placed in its middle at the pore. Polymer tends to minimize its
energy and is favorable to translocate to either side. In general, the
presence of $n^{*}$ indicates that translocation resembles a nucleation
phenomenon Muthukumar (1999). If the number of segments is less than $n^{*}$,
then segments still tend to return to donor side. The translocation process is
stochastic, like a nucleation phenomenon. Once a sufficient number of monomers
have crossed the nucleation barrier to the receiver side and are larger than
$n^{*}$, translocation to this side is more favorable as the free-energy
profile is now downhill. Hence, a critical number of $n^{*}$ ought to be
nucleated in the receiver side to make translocation successful. The variation
of free energy $F(n)$ with the entire length of the polymer with its critical
points represented with circles. This free energy curve is helpful in
understanding Fig. 5. In this plot, the probability of overcoming the
nucleation barrier for further successful translocation for asymmetrically
placed polymer vs. the number of monomers being shifted to the trans side is
plotted. Here, $N_{trans}$ is the length of the polymer on the trans side, and
$(N-1)/2$ represents the middle of the polymer $N$. Asymmetry introduced in
the polymer, which was initially placed symmetrically at the pore, is
represented in terms of the number of monomers shifted from its middle to the
trans side and shown as $N_{trans}-(N-1)/2$. From the probability plot, it can
be seen that as soon as we start placing the polymer asymmetrically on the
trans side, that is when it deviates from $N_{trans}-(N-1)/2=0$, which gives
$N_{trans}=(N-1)/2$ symmetrically placed polymer, it starts preferring the
trans (receiver) side. Once a significant section of the polymer is shifted to
the trans (receiver) side (say approximately $60\%$ of the length) and once
the nucleation barrier is crossed, the polymer slowly reaches a free energy
minima and it can undergo further translocation to receiver side without
coming back to the donor side.

In Fig. 7 we construct phase plots for $n^{*}/N$ in the $\Delta\mu-N$ plane.
The effect of the chemical potential gradient can be seen in Fig. 7. For a
crowd-free environment, $\Delta\mu=0$, $n^{*}/N$ is exactly half for all chain
lengths $N$, and the free energy barrier is symmetric, shown by the black
dotted line in Fig. 7. Another situation is translocation into a crowded
environment driven by a chemical potential gradient. For the case of one side
crowding where the trans side has crowders having $\phi_{c}=0$ and
$\Delta\mu=\mu_{cis}-\mu_{trans}=0-\mu=-\mu$, where $\mu$ is a positive
quantity and the polymer tends to move to the free cis side. However, a
critical value of polymer segment $n^{*}$ can reverse this trend, which is
sufficient to overcome the osmotic pressure. See the upper panel of Fig. 7 and
Fig. 8. The free energy barrier is bigger when the chemical potential gradient
$\Delta\mu=-\mu$ is in the opposite direction to the translocation process.
Since the barrier is bigger despite being crossed, the chain in the receiver
region remains in a metastable state and will eventually tend to revert to the
donor region. For this case, a higher value of $n^{*}/N$ is needed for a
favorable translocation in the receiver region. For lower pannel of Fig. 7 and
large values of $N$ and $\Delta\mu$, $n^{*}$ becomes progressively small. We
require only a small segment of the polymer to be shifted at the cost of
chemical potentials, such as for $N=100$, $\Delta\mu=0.01$, it can be
compensated by $n^{*}=22$ for the nucleation process to happen Muthukumar
(2016). In our model of polymer translocating through the crowd-free
environment, we have observed that nucleation phenomenons start at $n^{*}=10$.
The dependence of Free energy maxima $F^{*}$ on the chain length $N$ and
chemical potential gradient $\Delta\mu$ .

In Fig. 8, we can see a sudden drop in the curve when $\Delta\mu$ deviates
from zero. As soon as the crowding is introduced, starting from the small size
of the crowders, there is a critical value of $\sigma$ for a change in
$\Delta\mu$ at which the polymer chain will face energy downhill with respect
to $n$. This trend can be seen in the one-sided crowding case (Fig. 10). For
small $\sigma_{c}$ we have a high value of $\Delta\mu$, crowding has a strong
effect on pushing the polymer to the crowd-free side and resulting in a sudden
rise which gives $P_{c}=1$. While the case of higher $\sigma_{c}$ and
comparatively lower $\Delta\mu$ will create larger voids in the crowding side.
The polymer still has a small but non-zero $P_{c}$. This would suggest the
critical point where the translocation can act as the nucleation phenomenon,
with few monomers passing through the pore. Very less shifting of polymer,
even with the small value of $n$, will escalate the translocation to the
crowded side. A detailed discussion of the effect of one-sided (trans)
crowders on the translocation probability is done in the next section. The
dependence of this critical number $n*$ per unit length $N$ with changing
$\Delta\mu$ .

###  III.2 Effect of one-side crowders

Figure 9:  Schematic illustration of translocation process of a polymer in the
presence of crowder on the trans side. (a) The packing fraction of the crowder
on the trans side $\phi_{t}=0.3$ for $\sigma_{t}$ = 0.6 (case: $\sigma_{t}<1$)
(b)Packing fraction of the crowder on the trans side is $\phi_{t}=0.3$ for
$\sigma_{t}=4$ (case: $\sigma_{t}>1$). Figure 10: Upper panel: Translocation
probability as a function of packing fraction of inert crowders on the trans
side $\phi_{t}$. (a) Translocation probability $P_{c}$ to cis-side for
different $\sigma_{t}$. (b) Translocation probability $P_{t}$ to trans-side
for different $\sigma_{t}$. Lower panel: Translocation probability as a
function of $\sigma_{t}$ of inert crowders on the trans side. (c)
Translocation probability $P_{c}$ to cis-side for different $\phi_{t}$. (d)
Translocation probability $P_{t}$ to trans-side for different $\phi_{t}$.

We now study the impact of one-sided inert crowders on the translocation
process (Fig. 1b). The study of inert crowders on both sides Chen and Luo
(2013) and non-inert crowders on one side (chaperones) Yu and Luo (2011) has
been studied. However, a detailed investigation of translocation processes
with inert crowders on one side did not get its due attention. For this
purpose, we choose to put inert crowders only on one side, say the trans side,
in our case (Fig. 9) and studied the translocation phenomena of a
symmetrically placed flexible polymer. As expected, we have observed that the
translocation probability and time have a strong dependency on the packing
fraction $\phi_{t}$ and crowding size $\sigma_{t}$. To understand the
underlying mechanism of this effect, we first quantify the ballpark average
distance between the crowders $\langle r_{s}\rangle$ and mean entropic force
$\langle F_{r}\rangle$.

Consider $N_{t}$ is the total number of crowders on the trans-side, each with
size, $\sigma_{t}$ Chen and Luo (2013). The packing fraction on the trans-
side, with dimension $L_{x}\times L_{y}$ is

$\phi=\frac{N_{t}~{}\pi(\frac{\sigma_{t}}{2})^{2}}{L_{x}\times L_{y}}.$ (9)

Note that the dimension of the box is $2L_{x}\times L_{y}$. Now for a fixed
$\phi$, the average distance between crowders, taken from their center
$\langle r_{c}\rangle$ can be written as,

$\langle r_{c}\rangle=\frac{\sqrt{L_{x}\times L_{y}}}{\sqrt{N_{t}}}.$ (10)

Hence, the average distance between the surface of crowders $\langle
r_{s}\rangle$ is

$\langle r_{s}\rangle=\langle r_{c}\rangle-\sigma_{t}=\frac{\sqrt{L_{x}\times
L_{y}}}{\sqrt{N_{t}}}-\sigma_{t}.$ (11)

For the box of the same dimension, inserting the value Eq. 9 in the Eq. 11, we
get

$\langle r_{s}\rangle=\sigma_{t}\left(\frac{1}{2}\sqrt{\pi/\phi}-1\right).$
(12)

From the above expression we observed that the average distance between the
surface of the crowders $\langle r_{s}\rangle$, that is the space available
for the polymer to explore, is proportional to the crowder size, $\langle
r_{s}\rangle\propto\sigma_{t}$, and inversely proportional to the square-root
of the packing fraction, $\langle r_{s}\rangle\propto 1/\sqrt{\phi}$. This
behavior plays a crucial role in understanding the effect of crowders on
translocation. A detailed explanation of its significance can be found in the
coming section of translocation probability.

The focus of our study is on the regime of packing fraction of crowder
$\phi<0.5$. According to the percolation theory, the fluid phase is considered
continuous Stauffer and Aharony (2018); Essam (1980), indicating the formation
of channels by crowders.

Now, we consider average radial entropic force $\langle F_{r}\rangle$ and its
role in the translocation process Muthukumar and Baumgärtner (1989); Roos
(2014). Radial entropic force describes the force exerted on a monomer by the
particles as it moves through the channel formed by the crowders and drives
the polymer to translocate through the pore. It is proportional to the
temperature, inversely proportional to the confinement created by the
crowders, and equivalent to inter-crowders separation Sheng and Luo (2012);
Chen and Luo (2013). Mean radial entropic force can be written as

$\langle F_{r}\rangle=F_{0}\frac{k_{B}T}{\langle r_{s}\rangle},$ (13)

where $k_{B}$ is the Boltzmann constant, $T$ is the absolute temperature, and
$F_{0}$ is the proportionality constant depends on the solvent-polymer
interactions. Simplifying the expression in the terms of $\sigma$, $\phi$, we
have

$\langle
F_{r}\rangle=F_{0}\frac{k_{B}T}{\sigma(\frac{1}{2}\sqrt{\pi/\phi}-1)}$ (14)

The presence of crowders on the trans-side restricts the motion of the polymer
and reduces its possible configurational entropy, which in turn reduces the
entropic force acting on the polymer. This resulted in a lower entropic state
in the trans-side compared to the cis-side, which has no crowders and,
therefore, less restriction on the polymer’s motion. The difference in entropy
between the two sides serves as a driving force for the polymer to translocate
from the lower entropy (trans side) to the higher entropy (cis side) Chen and
Luo (2013).

To elucidate the dynamics of polymer translocation through a crowded
environment driven by different sizes, we examine the role of osmotic pressure
$(\Pi)$. Being a colligative property, osmotic pressure is proportional to the
solute concentration $\rho_{c}$, and temperature $T$, can be written as
$\Pi=R\rho_{c}T$, where $R$ is the constant of proportionality. It can be used
to determine the direction of polymer movement through the pore in response to
the difference in crowding concentration gradient on either sideDoi  _et al._
(1988); De Gennes and Gennes (1979); Muthukumar (2016). The polymer is
influenced by this osmotic pressure difference, which drives it to the cis-
side where the concentration and, hence, pressure are lower and have more free
volume. The entropic force also pulls the polymer in the direction of higher
entropy, which is towards the cis-side. The interplay between these two forces
determines the overall direction of polymer translocation through the pore.

####  III.2.1 Translocation probability as a function of $\phi_{t}$,
$\sigma_{t}$

Translocation probability to the trans-side $P_{t}$ is defined as the ratio of
successful translocation events towards the trans-side (right) to the total
number of successful translocation events that have occurred. The probability
of the polymer moving to the free cis-side (left) is denoted by $P_{c}$ and
note for this study of one-sided crowding, we always set $\phi_{c}=0$,
representing the free environment.

Fig. 10a shows the probability of the polymer to translocate to the cis-side
($P_{c}$) as a function of the packing fraction $\phi_{t}$, for different
values of $\sigma_{t}$. With increasing $\phi_{t}$, $P_{c}$ rapidly increases
($P_{c}=P_{t}=0.5$ at $\phi_{c}=\phi_{t}$=0) and eventually approaches
saturation ($P_{c}=1$). This indicates that, for the difference in crowding
concentration on either side, the polymer prefers moving to the free cis-side.
It can be understood on the basis of entropic ground and osmotic pressure.
Cis-side has a more free volume, which drives the polymer to translocate to
the free side to achieve a higher entropy state by minimizing the crowding-
induced constrained configurations. Additionally, the concentration gradient
of the crowders also plays an important role in pushing the polymer from the
crowded trans-side to the free cis-side. Whereas for $P_{t}$ varying with
$\phi_{t}$ at a fixed $\sigma_{t}$ shows complementary decrease
($P_{t}=1-P_{c}$) in its value (Fig. 10b). This also manifests the same effect
of entropy and osmotic pressure.

In Fig. 10c, d we represent how $P_{c}$ and $P_{t}$ varies with $\sigma_{t}$
for different values of $\phi_{t}$. We observe that $P_{c}$ decreases and
$P_{t}$ increases monotonically with increasing $\sigma_{t}$ for different
$\phi_{t}$. This trend continues for the lowest $\phi_{t}$ until it reaches
the equal probable case $P_{t}\approx P_{c}\approx 0.5$ at maximum
$\sigma_{t}$. While with increasing $\phi_{t}$, and for lower values of
$\sigma_{t}$, the curve rises slowly. Correspondingly for the highest
$\phi_{t}$, $P_{t}$ starts increasing only after $\sigma_{t}>3$, indicating a
need for a higher $\sigma_{t}$ to reach $P_{t}\neq 0$ at higher $\phi_{t}$.
The reason is that for a fixed $\phi_{t}$ on increasing $\sigma_{t}$, the
number of crowders decreases, thereby leading to higher $\langle r_{s}\rangle$
between the randomly distributed crowders, resulting in the existence of large
voids on the trans side that a polymer can explore and thereby increasing the
tendency of the polymer to translocate to the crowders side. Additionally,
this behavior simply corresponds to having effective osmotic pressure from the
crowders on the trans-side. The concentration gradient created by crowders
from the trans-side to the cis-side, that is, from higher osmotic pressure to
the lower one, also drives the polymer to move to the free cis-side. For the
highest packing fraction of the crowders $\phi=0.3$, with increasing
$\sigma_{t}$, crowders have more large voids resulting in more free volume for
the polymer to explore and also osmotic pressure at the trans-side is
comparatively less by higher $\sigma_{t}$ (lower $\rho_{c}$) than that by
lower $\sigma_{t}$ (higher $\rho_{c}$) and polymer prefers to stay towards the
crowder side which is reflected in $P_{t}$ curve.

In Fig. 11 we construct phase plots for $P_{c}$ and $P_{t}$ in the
$\phi_{t}-\sigma_{t}$ plane. $P_{c}$ is maximum for higher $\phi_{t}$ and
relatively smaller crowders (see Fig. 11a, $\phi_{t}>0.1$, $\sigma_{t}<1$),
indicating translocation towards the free-cis side occurs more readily in a
densely crowded medium with smaller crowding particles. While for the case of
bigger crowders size $\sigma_{t}$, the effect of crowders to push the polymer
to the free-cis side decreases because of the presence of larger voids in the
trans side and probability of translocation to the trans side increases
$P_{t}>0.$ Fig. 11b is the complimentary phase plot of Fig. 11a.

####  III.2.2 Translocation time as a function of $\phi$, $\sigma_{t}$

Now, we study the effect of packing fraction and crowder size on translocation
time. Here, the system of interest consists of both polymer and crowder on the
trans-side. Both entities owe different time scales. In 2D, for a crowder
moving with a diffusion constant $D_{0}=\frac{K_{B}T}{\xi}$, the time scale
associated with the movement to a distance of the order of their size $\sigma$
is

$\tau_{0}=\frac{\sigma^{2}}{4D_{0}}=\frac{\sigma^{2}\xi}{4k_{B}T},$ (15)

where $k_{B}$ is the Boltzamnn constant and $T$ is absolute temperature. For
the bigger crowder of diameter $\sigma_{t}$, time is $\tau$

$\tau=\frac{\sigma_{t}^{2}}{4D}.$ (16)

Therefore,

$\tau=\tau_{0}\frac{\sigma_{t}^{2}}{\sigma_{0}^{2}}.$ (17)

It shows that the time scale for diffusive motion is proportional to the
damping constant and the size of the crowder. For same $\xi$, the $\tau$
increases rapidly on increasing $\sigma_{t}$ Chen and Luo (2013).

As per the Rouse model, the polymer is represented as a chain having N beads.
The diffusion coefficient of the Rouse chain is obtained by Einstein’s
relation $D_{R}=\frac{k_{B}T}{\xi_{R}}$. The polymer diffuses a distance of
the order of its size during a characteristic time, called Rouse time
$\tau_{R}$ is Doi  _et al._ (1988); Rubinstein  _et al._ (2003); Chen and Luo
(2013)

$\tau_{R}=\tau_{0}N^{(1+2\nu)},$ (18)

where $\nu$ is the flory exponent. As polymers exhibit self-similarity, they
are characterized by $N$ distinct relaxation modes. Each relaxation mode is
designated by a mode index $p=1,2,3,\cdots,N$ can be written as

$\tau_{p}=\tau_{0}\frac{N}{p}N^{(1+2\nu)}.$ (19)

Within the system, the polymer is a large object that moves slowly, while the
crowder movement depends on their size. In cases where the time scales for the
crowder is shorter than $\tau_{p}=\tau_{0}$ for $p=N$, the correlation between
the crowders is insignificant, and they provide a uniform random background
for the polymer. However, when the two-time scales are comparable, as in this
study, the polymer’s motion correlates with the crowder’s. As the size of the
crowder increases, their correlations become stronger. On increasing the
packing fraction of crowders, the diffusion coefficient of polymer and crowder
decreases, which means correlations between the crowders become stronger.

$\tau_{t}$ and $\tau_{c}$ is the time the polymer takes to translocate to the
crowded trans side and to the free cis-side. In Fig. 12a and 12b we have
plotted $\tau_{c}$ and $\tau_{t}$ as a function of $\phi_{t}$ for different
$\sigma_{t}$. $\tau_{c}$ decreases monotonically with increasing $\phi$. A
decrease in overall $\tau_{c}$ can be understood in terms of mean entropic
force. On this basis, an increase in $\phi_{t}$ and random distribution of
crowders on the trans-side will lead to lower entropy on this side and drive
the polymer to translocate to the side of no crowder having higher entropy.
Hence, in less time, $\tau_{c}$ as compared to $\tau_{t}$ polymer gets
translocated to no crowding side. A sudden jump in the graph for higher
$\sigma_{t}$ can be interpreted from Fig. 12c, d. It shows that translocation
time is a function of $\sigma_{t}$ for different $\phi_{t}$. $\tau_{c}$
increases with an increase in $\sigma_{t}$ and a decrease in $\phi_{t}$. The
increase in $\tau_{t}$ is more rapid as compared to the $\tau_{c}$. For a
fixed $\phi_{t}$, on larger $\sigma_{t}$, the case of translocation to trans-
side becomes less probable, leading to higher translocation time.
Specifically, for higher $\sigma_{t}$, that is, $\sigma_{t}\geq 3$, the
increase in the size of crowders on the trans-side leads to a rare, large void
and more space for the polymer to move. Hence, the $\tau_{t}$ shows a sudden
increase in its value. Also, in terms of osmotic pressure, as the number of
crowders increases, the force exerted by them on the polymer also increases.
For a fixed $\phi_{t}$ and the fixed area of the box on the trans-side, the
$N_{t}$ has a direct relation with the force exerted by them on the polymer
and an inverse relation with the square of the size of the crowder. This also
manifests the sudden jump in $\tau_{t}$ behavior (Fig. 12d). Unlike Kaifu and
Luo et al. Chen and Luo (2013), there is no resistive force on the polymer
translocation (even in the case of one side free crowders where the polymer
has more entropy to go to the cis-side but prefers trans-side having bigger
crowder) in fact polymer translocation to trans-side is effectively increased
for higher $\sigma_{t}$.

Figure 11:  The plot elucidates how translocation probabilities $P_{c}$ and
$P_{t}$ vary as the function of the size of crowders ($\sigma_{t}$) and their
packing fraction ($\phi_{t}$) in one-sided (trans) crowded environment. (a)
Translocation probability $P_{c}$ to the free-side (cis). (b) Translocation
probability $P_{t}$ to the crowded side (trans).  Figure 12: Upper Panel:
Translocation time as a function of packing fraction of inert crowders
$\phi_{t}$ on the trans-side. (a) Translocation time $\tau_{c}$ to the free
cis-side for different $\sigma_{t}$. (b) Translocation time $\tau_{t}$ to the
crowded trans-side for different $\sigma_{t}$. Lower Panel: Translocation time
as a function of size of inert crowders $\sigma_{t}$ on the trans-side. (c)
Translocation time $\tau_{c}$ to the free cis-side for different $\phi_{t}$.
(d) Translocation time $\tau_{t}$ to the crowded trans-side for different
$\phi_{t}$.

####  III.2.3 Non zero translocation probability towards the crowders side

Figure 13: Sigma splitting. Packing fraction of the crowder on trans-side
$\phi_{t}$ as a function of critical value of sigma
$\sigma_{t}\approx\sigma^{*}$, that is, the size of crowders above which
polymer have non-zero $P_{t}$.

It is expected for one-sided crowders that translocation to the free cis-side
will always be preferred. For small crowding sizes, when the effect of the
packing fraction is the strongest, there is almost $100\%$ certainty that the
polymer will translocate to the cis side, leaving $P_{t}=0$. However, we have
already discussed that as we increase the crowding size $\sigma_{t}$ for the
same $\phi$, the effective impact of the crowding environment weakens and
gives rise to non-zero $P_{t}$. For any specific value of $\phi$, there is
minimum $\sigma_{t}$, say $\sigma^{*}$, from where $P_{t}$ starts to become
$P_{t}>0$. This minimum value of $\sigma_{t}$, we define as critical size
$\sigma^{*}$ (Fig. 13). It will be interesting to observe how this critical
value of crowder size $\sigma^{*}$ depends on the packing fraction $\phi$.
Note from Fig. 10c that these $\sigma^{*}$, the critical values of
$\sigma_{t}$, increase with $\phi_{t}$ and follows an almost linear relation,
$\phi_{t}\approx\sigma^{*}$. This can be understood on the basis of entropy
where the voids created on crowded trans-side provide a large room and hence
higher entropy for the polymer to explore than from cis-side, which results in
a non-zero $P_{t}$.

####  III.2.4  Effect of asymmetric initial configuration of polymer on
translocation probability

For the initial part of our study, we investigated the translocation dynamics
of a polymer through a pore in the presence of free and one-sided crowder.
Specifically, we have examined the dynamics of the polymer when it passes with
its middle monomer at the pore between cis and trans side of the box. In the
free case, when there are no crowders, the environment is symmetric, and a
symmetrically placed polymer with a middle monomer at the pore exhibits an
equal probability of translocation to either side ($P_{c}=P_{t}=0.5$). The
introduction of crowders on one side of the box breaks the symmetry in the
translocation dynamics of the polymer, resulting in a preference for
translocation in the direction of lower osmotic pressure and higher entropy
($P_{c}=1,P_{t}=0$)Bhattacharjee  _et al._ (2013); Gopinathan and Kim (2007);
Roos (2014). Now, we look at how we can overcome the translocation barrier
imposed by the crowders by introducing asymmetry to its length. To restore the
symmetric behavior observed in the absence of crowders, we positioned the
polymer asymmetrically on the side with crowders and provided it with an extra
shifted length. Our study suggests that the additional length of the polymer
provides an edge to the crowders, allowing them to bias the direction of
translocation more effectively. Fig. 14 represents the translocation
probability $P$ to either side with the ratio of shifted length $L_{t}$ of the
polymer of length $L$. The first case is a polymer shifting from a sparsely
crowded environment of $\phi$= 0.1 (Fig. 14a) with different lengths to a
densely packed crowding $\phi$ = 0.2 (Fig. 14b). As the polymer shifts to the
trans-side where crowders reside, the likelihood of the polymer crossing over
to the free cis-side begins to decrease from its maximum value. This decrease
is due to a reduction in the length of the polymer on the cis-side, resulting
in a decrease in available conformational entropy. A channel exists in the
vicinity of the pore. The cylindrical nanochannel created by the crowders
provides confinement to the polymer, resulting in a lower available area for
the polymer to dangle freely. As we increase the asymmetry by shifting the
polymer more to the crowder side, the confinement produced by the crowders in
the pore’s proximity constrains the motion on the free side due to propagation
through the backbone of the polymer. Once an interesting situation arises
where the polymer on the crowder side has more freedom to move compared to the
confined part of the channel on the cis -side, which gives the polymer the
push to move towards the crowder side. Howsoever, the length of the polymer,
this channel will compensate for its effect on either side of the pore. This
can also be interpreted in terms of entropy, the trans side being crowded and
having more monomers due to shifted length by which configurational entropy
increases; hence, the driving force for the polymer to move towards the
crowders side increases. Consequently, there is an increase in the $P_{t}$
value. The two curves converge at a shifted length of $L_{t}/L$, where the
probability of translocation to either side is $50\%$, making the polymer
unbiased. On shifting polymer length towards the crowders, they create a
higher resistance for the polymer as it tries to translocate through the pore.
As the crowding level increases for higher $\phi_{t}$ (Fig. 14b), the polymer
experiences more resistance, and the probability of translocating towards
either side becomes balanced at higher shifted length. A comparative plot for
both packing fractions comprising of shifted length against the total length
of the polymer is shown in Fig. 15.

On the crowded trans side, the free energy equation remains F=E-TS but with a
substantial positive E value (repulsive interactions with crowders and
polymer). This results in higher free energy compared to the cis side, leading
to lower entropy on the trans side. Introducing asymmetry by shifting the
polymer to the trans side increases the number of monomers on the trans side,
elevating the interaction energy contribution for the shifted length.
Simultaneously, the shifted polymer enhances the configurational entropy on
the trans side compared to the dangling tail on the cis side. The net effect
is a decrease in the free energy of the chain and higher entropy on the trans
side, facilitating translocation toward the crowded trans side.

Figure 14: Shifing of polymer length in the crowded trans-side.(a) Sparse
crowding $\phi_{t}=0.1$ (b) Comparatively dense crowding $\phi_{t}=0.2$ with
polymer length, N = 43, 65, 91, and 101. Figure 15: Combined plot of polymer
length shifted $L_{t}/L$ to crowded environments for two different
$\phi_{t}=0.1,0.2$ against the original length of the polymer $L$. Both the
plots show consistent behavior for a particular value of $\phi_{t}$ as a
function N. Figure 16: Schematic illustration of polymer translocation process
through a pore with asymmetric crowders on both sides of the box. The size of
crowders on the cis-side is kept fixed $\sigma_{c}=1$. The size of crowders on
trans-side $\sigma_{t}$ keeps on changing while keeping the packing fraction
on both sides alike ($\phi_{t}=\phi_{c}=\phi$). (a) Packing fraction of
crowders on both sides $\phi=0.3$ (densely crowded) while $\sigma_{c}=1$ and
$\sigma_{t}=0.6$ (case: $\sigma_{t}<\sigma_{c}$). (b) Packing fraction of
crowders on both sides $\phi=0.3$ while $\sigma_{c}$ = 1 and $\sigma_{t}$ = 4
(case: $\sigma_{t}>\sigma_{c}$)

###  III.3 Effect of both sided crowders

Now, we study the translocation dynamics of a flexible polymer driven by both
side crowders to mimic environments similar to biological situations and in-
vitro setups (Fig. 1c). In this study, we kept both side packing fractions
equal: $\phi_{c}=\phi_{t}$ and varied the relative size of the crowders. For
simplicity, we always kept the cis-side crowding size fixed at $\sigma_{c}=1$
and varied the trans-side crowding size $\sigma_{t}$ (Fig. 16). To start, we
positioned the middle monomer ($N/2+1$) of the polymer chain at the pore and
allowed the beads in the chain to reach equilibrium conformations through
thermal collisions. Once the equilibration is completed, the polymer achieves
a random configurational state, which is considered to be the initial
confirmation for the translocation process. A successful translocation
occurred when the polymer chain ended up on either side of the pore within the
simulation time. For successful translocations, we have observed translocation
probability and time as a function of packing fraction $\phi$ and crowding
size $\sigma_{t}$. This phenomena of polymer translocation driven by crowders
can be elucidated by quantifying the average distance among the crowders
$\langle r_{s}\rangle$ and mean entropic force $\langle F_{r}\rangle$.

Consider that there are $N_{c}$ crowders in general of the diameter
$\sigma_{i}$ on either side of the box with dimension ($L_{x}\times L_{y}$),
then the packing fraction is

$\phi=\frac{N_{c}~{}\pi(\frac{\sigma_{i}}{2})^{2}}{L_{x}\times L_{y}}.$ (20)

Using Eq.(17) and alike Eq.(7), the average distance between the surface of
the crowders $\langle r_{s}\rangle$ is

$\langle r_{s}\rangle=\sigma_{i}\left(\frac{1}{2}\sqrt{\pi/\phi}-1\right),$
(21)

The packing fraction of crowders at both sides is set as
$\phi_{t}=\phi_{c}=\phi$, and the size of the crowder at the cis-side is kept
fixed $\sigma_{i}=\sigma_{c}=1$. On the trans side, the size of the crowder
varies $\sigma_{t}=\sigma_{i}$ within $0.6\leq\sigma_{t}\leq 2.5$ for our
case. Thus, the average distance between the surface of the cis-side crowders
$\langle r_{cs}\rangle$ and trans-side crowder $\langle r_{ts}\rangle$ is

$\langle
r_{cs}\rangle=\sigma_{c}\left(\frac{1}{2}\sqrt{\pi/\phi}-1\right),~{}~{}~{}\langle
r_{ts}\rangle=\sigma_{t}\left(\frac{1}{2}\sqrt{\pi/\phi}-1\right).$ (22)

Eq. 21 and Eq. 22 indicates that for a fixed $\phi$, $\langle r_{s}\rangle$
increase with increasing $\sigma_{i}$ and it decreases with increasing $\phi$
for a fixed $\sigma_{i}$. It will provide insight into understanding the
phenomena of translocation induced by crowders and the average distance
between their surfaces.

As the polymer is coming out of a pore, crowders create a confined cylindrical
channel of size $\langle r_{s}\rangle$. For this mean entropic force $\langle
F_{r}\rangle$ can be written asRoos (2014); Chen and Luo (2013); Muthukumar
and Baumgärtner (1989)

$\langle F_{r}\rangle=F_{0}\frac{k_{B}T}{\langle r_{s}\rangle},$ (23)

where $F_{0}$ is the proportionality constant, $k_{B}$ is the Boltzmann
constant, $T$ is the absolute temperature.

Simplifying the expression of $\langle F_{r}\rangle$ for cis and trans-side
crowder in terms of $\langle r_{cs}\rangle$, $\langle r_{ts}\rangle$,
$\sigma_{c}$, $\sigma_{t}$, and $\phi$ gives

$\langle F_{r}\rangle=F_{0}k_{B}T\left(\frac{1}{\langle
r_{cs}\rangle}-\frac{1}{\langle r_{ts}\rangle}\right),$ (24) $\langle
F_{r}\rangle=F_{0}k_{B}T{\frac{1}{\frac{1}{2}\sqrt{\pi/\phi}-1}\left(\frac{1}{\sigma_{c}}-\frac{1}{\sigma_{t}}\right).}$
(25)

Eq. 25 show that $\langle F_{r}\rangle$ depends on $\phi$ and varying trans-
side crowders of $\sigma_{t}$. With increasing $\phi$ or changing $\sigma_{t}$
keeping $\sigma_{c}$ fixed, $\langle F_{r}\rangle$ increases Chen and Luo
(2013). Polymer prefers to move the size of bigger crowders due to larger
conformational entropy.

Further, Osmotic pressure ($\Pi$), which is a colligative property, depends on
the number density of the crowder and influences the translocation dynamics.
We put a polymer chain at the center of the wall, separated by a difference in
crowding size, which creates a crowding concentration gradient on either side.
The polymer chain will be driven by the difference in osmotic pressure created
by the concentration gradient and prefers to stay on the side of relatively
lower osmotic pressure and higher volume. Thus, the competition between mean
entropic force and osmotic pressure determines the direction of the
translocation process.

####  III.3.1 Translocation probability as a function of $\phi$, and
$\sigma_{t}$

Figure 17: Upper panel: Translocation probability as a function of packing
fraction $\phi$. (a) Translocation probability $P_{c}$ to cis-side for
different $\sigma_{t}$. (b) Translocation probability $P_{t}$ to trans-side
for different $\sigma_{t}$. Lower panel: Translocation probability as a
function of $\sigma_{t}$ of inert crowders on the trans side. (c)
Translocation probability $P_{c}$ to cis-side for different $\phi_{t}$. (d)
Translocation probability $P_{t}$ to trans-side for different $\phi_{t}$.
Figure 18: The plot elucidates how translocation probabilities $P_{c}$ and
$P_{t}$ vary as the function of the size of crowders on the trans side
($\sigma_{t}$) and their packing fraction ($\phi$) in a both-sided crowded
environment. The size of crowders on the cis side is kept fixed at
$\sigma_{c}=1$, and the packing fraction of both sides is kept equal, that is,
$\phi_{c}=\phi_{t}=\phi$. It shows a switch in the translocation of a polymer
by tuning crowding size. (a) and (b) Translocation probability to cis-side and
trans-side, respectively.

The probability of polymer translocating to the cis-side and trans-side is
represented by $P_{c}$ and $P_{t}$, respectively. In Fig 17a and Fig 17b, we
have plotted the translocation probability of the polymer as a function of the
packing fraction $\phi$ for different $\sigma_{t}$, note $\sigma_{c}$ is
always kept constant at $\sigma_{c}=1$. Plot shows bifurcation from equally
probable scenario ($P_{t}=P_{c}=0.5$ at $\sigma_{t}=1$) to two opposite
extremes ( $\sigma_{t}\neq\sigma_{c}$ ). For $\sigma_{t}\leq\sigma_{c}$, the
polymer has a tendency to move to the cis side where bigger crowder resides.
$P_{c}$ rapidly increases from 50-50 $\%$ at lower $\phi$ and then slowly
reaches its maximum value at higher $\phi$. While for the case of
$\sigma_{t}\geq\sigma_{c}$, the polymer has a tendency to move to the trans
side. This indicates that for the same $\phi$ on both sides, translocation is
preferred to the side of bigger crowders. This behavior is reminiscent of the
effect of osmotic pressure, which provides a push to the polymer to
translocate from a higher crowder concentration to a lower concentration side.
For the same $\phi$, osmotic pressure ($\Pi$) at the side of the smaller
crowder is higher than that of the side of a bigger one, leading to the
polymer moving to the bigger crowder side. Such behavior of probability with
$\phi$ for polymer translocation through pore have been reported earlier Chen
and Luo (2013).

We next look at the translocation probability of the polymer as a function of
$\sigma_{t}$ for different $\phi$. $P_{t}$ displays distinct features as
$\sigma_{t}$ is varied (Fig. 17c, d). For $\sigma_{t}<\sigma_{c}$ till it
reaches $\sigma_{t}=1$, that is, for the relatively small trans-side crowders,
polymer translocates to the cis-side ($P_{t}=0$). But, the moment there is a
crossover of $\sigma_{t}=1$, the curve shows a sudden jump in its behavior
from its lowest to saturation ($P_{t}=1$). This sudden switch prevails over
the higher $\sigma_{t}$ for the highest $\phi=0.3$ and shows a fall from
$P_{t}=1$ for decreasing $\phi$. Additionally, even for $\phi=0.01$, which is
almost a crowder-free environment, the strong effect of tiny crowders
($\sigma_{t}<\sigma_{c}$) pushes the polymer to translocate to the cis-side
($P_{t}=0.3$). While for the same $\phi=0.01$ at $\sigma_{t}>\sigma_{c}$, the
polymer has an equal probability of moving to either side as it is like a tug-
of-war situation. This result can be interpreted in terms of the mean radial
entropic force. $\langle F_{r}\rangle$ increases with increasing $\sigma_{t}$
and $\phi$ and hence leads the translocation dynamics. Unlike Chen et al.,
2013 Chen and Luo (2013), our model has no bottleneck or resistive force
pertaining to non-monotonic behavior.
The results in Fig. 17 are relevant to the simulation study by Chen and
LuoChen and Luo (2013). They observed that the polymer with an initial
configuration where the middle monomer at the pore on the wall separating two
sides will prefer to translocate to the side of bigger crowders. They note
that polymer that goes to the side of a bigger crowder exhibits a maximum
probability on increasing $\sigma$ up to a certain extent and then shows a
downfall due to the interplay of the driving force and the resistive force,
specifically from the bottleneck leading to the non-monotonic behavior of
probability. They observed that at large $\sigma$, it shows the resistive
force of large magnitude compensated by entropic force leads to a decrease in
probability to the bigger crowder side. While, in our case, the calculations
are consistent with Chen and Luo and likewise to Polson Polson and Heckbert
(2019), our model also contradicts this and denies any bottleneck in the
system pertaining to non-monotonicity in the curve. In our case, the
probability curve exhibits a continuous increase as the polymer prefers to
translocate to the side of bigger crowders and does not account for the
downfall in the curve is noted.

The behavior of the crossover region of $\sigma_{t}=1$ as a function of $\phi$
is presented in phase plot for $P_{c}$ and $P_{t}$ in $\phi-\sigma_{t}$ plane
(Fig. 18). $P_{c}$ is maximum for smaller $\sigma_{t}$ and higher $\phi_{t}$
(see Fig. 18a, $\sigma_{t}<1,\phi>0.05$). For relatively bigger crowders, the
phase plot shows a uniform increase in $P_{t}$ with increasing $\sigma_{t}$
for all $\phi$ (see Fig. 18b).

Figure 19:  Upper Panel: Translocation time as a function of packing fraction
$\phi_{t}$. (a) Translocation time $\tau_{c}$ to the cis-side for different
$\sigma_{t}$. (b) Translocation time $\tau_{t}$ to the trans-side for
different $\sigma_{t}$. Lower Panel: Translocation time as a function of size
of inert crowders $\sigma_{t}$ on the trans-side. (c) Translocation time
$\tau_{c}$ to the cis-side for different $\phi_{t}$. (d) Translocation time
$\tau_{t}$ to the trans-side for different $\phi_{t}$.

####  III.3.2 Translocation time as a function of $\phi$, and $\sigma_{t}$

We first look at the translocation time as a function of packing fraction
$\phi$ for different values of crowder size $\sigma_{t}$ (Fig. 19a, b). Owing
to an alike environment on both sides of the pore for
$\sigma_{t}=\sigma_{c}=1$, the overall $\tau$ owes the highest value. Further
diverging from $\sigma_{c}=\sigma_{t}=1$ and moving to the asymmetric nature
of crowded size ($\sigma_{c}\neq\sigma_{t}$), we observe a decrease $\tau_{c}$
and $\tau_{t}$ with increasing $\phi$ for both the situation:
$\sigma_{t}<\sigma_{c}$, and $\sigma_{t}>\sigma_{c}$. This result can be
elucidated in the context of $\langle F_{r}\rangle$, which increases on
increasing $\phi$. For $\sigma_{t}<\sigma_{c}$, the combined effect of osmotic
pressure $\Pi$ and mean entropic force $\langle F_{r}\rangle$ moves the
polymer to lower crowding concentration and higher entropy which dominate the
translocation phenomenology and drives the polymer on bigger crowder side of
the pore ensuing lower $\tau_{c}$ (Fig. 19a) and for $\sigma_{t}>\sigma_{c}$
holds the higher value of the average distance between the surface of crowder
on trans-side than on the cis side, providing more space for the polymer to
translocate easily in lesser time $\tau_{t}$ (Fig. 19b). Additionally,
translocation is a stochastic process, and random fluctuations could lead the
polymer to move to either the cis or trans side. But it’s not necessary that
translocation would occur for all particular cases, and pointed out
translocation at these points is a very rare event since we introduced biases
in the form of crowders in extreme bias, we wouldn’t observe any translocation
event taking place to the unfavoured side. In Fig. 19a, translocation time
values corresponding to $\sigma_{t}=0.8,2.0$ are not present at certain values
of $\phi$. This absence is attributed to the rare and stochastic nature of
translocation events. The lack of translocation time values on either side
indicates the non-occurrence of translocation in that specific direction. To
elaborate, when translocation doesn’t occur on the cis side at a given time,
but the polymer successfully translocates to the trans side, the corresponding
translocation time values are reflected in Fig. 19b. We next look at the
situation for translocation time varying with $\sigma_{t}$ for different
$\phi$ (Fig. 19c, d). At lower $\sigma_{t}<\sigma_{c}$, $\tau_{c}$ rises
faster for higher $\phi$. On the other hand for $\sigma_{t}>\sigma_{c}$,
$\tau_{t}$ shows decreasing behaviour on increasing $\phi$. As explained
earlier, the combined effect of $\Pi$, $\langle F_{r}\rangle$, $\langle
r_{s}\rangle$ pulls the polymer rapidly to the side of bigger crowder (Fig.
19c and Fig. 19d).

Figure 20: (a)Translocation probability $P$ as a function ratio of packing
fraction of crowders on the trans-side and cis side $\phi_{t}/\phi_{c}$ for
two distinct values of $\phi=0.1,0.2$. The size of crowder on the cis side
$\sigma_{c}=3$ while that on the trans side $\sigma_{t}=4$. (b) The size of
crowder on the cis side $\sigma_{c}=3$ while that on the trans side
$\sigma_{t}=2$.

####  III.3.3 Effect of crowding size on translocation probability

Translocation for the case of no crowders provides us the $50-50\%$
probability ($P_{c}\approx P_{t}\approx 0.5$). Initially, the symmetry of the
environment of obtaining equal probable case has been broken by introducing
crowders on one side ($P_{c}=1,~{}P_{t}=0$). We have observed that most of the
time polymer is translocating to the side of the crowders on the grounds of
mean entropic force, osmotic pressure, and distance between the surface of the
crowders. Next, the symmetry has been broken by introducing asymmetry in the
initial configuration of the polymer, that is, shifting the length towards the
crowder side. A ratio of $L_{t}/L$ exists where the equal probable case of cis
and trans-translocation is retrieved.

Now, when the crowders are being introduced on both sides of the box, we have
investigated that the polymer is translocating almost to the side of bigger
crowders ($P_{c}=0,P_{t}=1$ for $\sigma_{t}<\sigma_{c}$, and $P_{c}=1,P_{t}=0$
for $\sigma_{t}>\sigma_{c}$). Similarly, in this case, to restore an equal
probability scenario, symmetry is broken by changing the packing fraction of
crowders on the trans side, i.e., $\phi_{c}$ is not equal to $\phi_{t}$
anymore. We are choosing a particular value of the $\sigma_{t}$ and varying
the number of crowders $N_{t}$, which in turn changes the packing fraction
$\phi_{t}$. The goal is to obtain an optimum value of the packing fraction of
crowders on the trans-side so that the polymer translocates equally to both
sides rather than having a preference for the side of the bigger crowder. Fig.
20a and Fig. 20b illustrate the relationship between the probability of
translocation P and the ratio of packing fractions of two sides
$\phi_{t}/\phi_{c}$. In Fig. 20a, the crowders on the trans-side are larger
($\sigma_{c}=4$) than those on the cis-side ($\sigma_{c}=3$), while in Fig.
20b, the trans-side has smaller crowders ($\sigma_{c}=2$) than the cis-side
($\sigma_{c}=3$). Both figures examine sparse and densely crowded environments
of packing fraction on the cis-side ($\phi_{c}=0.1,0.2$), with
($\sigma_{c}=3$) held constant while $N_{c}$ is fixed. $N_{t}$ varies until
the polymer becomes unbiased of size, packing fraction, and the number of
crowders, allowing it to translocate to either side with equal probability.

##  IV Conclusion

Medium-driven controlled polymer translocations without any explicit external
forces are fundamental problems in science and engineering. Using Langevin
dynamics simulations, we extensively study the effect of crowding and polymer
length distribution on the translocation processes in crowd-free and crowded
environments. We investigated the scaling properties of translocation
probability $P$, time $\tau$, and MSD. First, we compared the scaling
properties with the results generated by the standard scaling theories of
polymer physics. We obtained scaling for translocation time $\tau\sim N^{2.5}$
and when $\tau>\tau_{R}$ MSD shows $\langle\Delta r^{2}(t)\rangle\sim
t^{0.8}$, in free environment. Our results exactly match the experimental data
and show subdiffusive behavior. A simple analysis of the average crossing time
$\langle\tau_{crossing}\rangle$ of individual monomer, as it crosses the pore,
has been performed and laid the foundation of two interesting but
counterintuitive features of translocation rate $k_{T}$ and bead velocity
$v_{b}$. It has been observed that the translocation rate is minimum when the
middle monomer passes the pore and keeps on increasing, whereas it holds the
opposite behavior for bead velocity, which is maximum for the middle monomer
and decreases for the subsequent part of tail monomers. Further, we
rationalize the translocation phenomena in the light of nucleation processes.
An asymmetrically placed polymer at the pore towards the receiver side once
crosses a significant number of monomers across the pore; that is, when the
nucleation barrier has been overcome and free energy maxima is reached, the
probability of the polymer going back to the donor side is almost negligible.

For one-sided crowding, with crowding on the trans side and free on the cis
side, the translocation probability and time have been noted. $P_{t}$ shows a
monotonic decrease with increasing $\phi$ but shows an increase when the
crowder’s size $\sigma_{t}$ is large enough to create rare, large voids the
polymer can explore. This provides us with a critical value of $\sigma_{t}$
called $\sigma^{*}$ where the polymer has non-zero $P_{t}$ and translocates to
the crowded side. A complimentary behavior is also followed by $P_{c}$. A
breakthrough in the existing model has been achieved by simply shifting
polymer length from the middle monomer to the crowded side to examine whether
there can be a situation if a crowd-free environment probability can be
retraced. The study has been done for both sparse environment $\phi_{t}=0.1$
and relatively dense $\phi_{t}=0.2$ and we investigated that there exists a
ratio of length shift to the polymer length $L_{t}/L$ where polymer exhibits
$P_{t}\approx P_{c}\approx 0.5$. Translocation time $\tau_{t}$ decreases
rapidly with increasing $\sigma_{t}$ and an increase with larger $\sigma_{t}$.

In the case of two-sided crowding, where the packing fraction of both sides is
set fixed, we kept the crowding side on the cis side $\sigma_{c}$ fixed, and
on the trans side, $\sigma_{t}$ is varying. The translocation probability
shows a bifurcation from the equal probable case, and polymer translocates to
the side of bigger crowders, $P_{t}>0.5$ for $\sigma_{t}>\sigma_{c}$ and
$P_{t}<0.5$ for $\sigma_{t}<\sigma_{c}$. The probability curve shows a sudden
switch from its lowest to peak value when a crossover of
$\sigma_{t}=\sigma_{c}$ happens. Translocation time $\tau_{t}$ decreases with
increasing $\phi$ and $\sigma_{t}$, indicating the polymer preference to stay
towards bigger crowders in less time. A study on changing the relative packing
fraction of crowders on both sides leads to the $50-50$ probability akin to
the crowd-free situation is done. These results can be interpreted by the
interplay between mean entropic force and effective osmotic pressure of the
crowders.

In the case of driven translocation, when an external force is applied on the
polymer up to its certain length and after that, if the force is switched off,
then the entropy-driven force and free energy are sufficient to keep the
polymer on the preferred side. Our model is generally enough to switch the
translocation direction without any explicit force. Once the nucleation
barrier is crossed for successful translocation, the polymer has an almost
extremely low probability of going back to the donor side. In other words, the
regulation of the free energy barrier by tuning the polymer length can have
significance in the translocation phenomena having biomedical applications,
for example, in the field of controlled drug delivery systems. Further, both
spatial and temporal control over the translocation of a polymer tethered to a
surface can be incorporated into our model. In the context of targeted drug
delivery systems, we aim to deliver drugs specifically to designated cells,
organelles, etc. This is the direct and potential application of polymer
translocation, allowing us to control the direction of polymer translocation
in accordance with the target’s environment Bai  _et al._ (2020); Zhang  _et
al._ (2012); Maitra Roy  _et al._ (2023). Controlled translocation of single
molecules can be achieved by maintaining control over the nanopore in such a
way that we have a speed tuning of individual beads and molecules as well as
passing through it at the desired time; that is, molecule-independent speed
control can be achieved Leitao  _et al._ (2023).

###### Acknowledgements.

We would like to thank Jaeoh Shin, Praveen Bommineni, Kunal Rai, and
Thyageshwar Chandran for useful discussion and critical comments. VG and SKG
would like to thank SERB, DST (Project No. SRG/2020/001606 ) for funding. SKG
would like to thank SERB, DST (Project No. MTR/2022/000833) for financial
support. We would like to acknowledge NIT Warangal for the research seed funds
and institute fellowship to VG.

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