] Author to whom the correspondence should be addressed

# DNA Barcodes using a Cylindrical Nanopore

Swarnadeep Seth  Aniket Bhattacharya  [<EMAIL_ADDRESS>1Department of Physics, University of Central Florida, Orlando, Florida
32816-2385, USA

###### Abstract

We report an accurate method to determine DNA barcodes from the dwell time
measurement of protein tags (barcodes) along the DNA backbone using Brownian
dynamics simulation of a model DNA and use a recursive theoretical scheme
which improves the measurements to almost 100% accuracy. The heavier protein
tags along the DNA backbone introduce a large speed variation in the chain
that can be understood using the idea of non-equilibrium tension propagation
theory. However, from an initial rough characterization of velocities into
“fast” (nucleotides) and “slow” (protein tags) domains, we introduce a
physically motivated interpolation scheme that enables us to determine the
barcode velocities rather accurately. Our theoretical analysis of the motion
of the DNA through a cylindrical nanopore opens up the possibility of its
experimental realization and carries over to multi-nanopore devices used for
barcoding.

A DNA barcode consists of a short strand of DNA sequence taken from a targeted
gene like COI or cox I (Cytochrome C Oxidase 1) [1] present in the
mitochondrial gene in animals. The unique combination of nucleotide bases in
barcode allows us to distinguish one species from another. Unlike relying on
the traditional taxonomical identification methods, DNA barcoding provides an
alternative and reliable framework to categorize a wide variety of specimens
obtained from the natural environment. Though researchers relied on DNA
sequencing techniques for the identification of unknown species for a long
time, in 2003, Hebert et al. [2] proposed the mictocondrial gene (COI) region
barcoding to classify cryptic species [3] from the entire animal population.
Since then, several studies have shown the potential applications of barcoding
in conserving biodiversity [4], estimating phyletic diversity, identifying
disease vectors [5], authenticating herbal products [6], unambiguously
labeling the food products [7, 8], and protecting endangered species [4].
Traditional sequencing methods based on chemical analysis are widely used in
the biological community to determine the barcodes. Nanopore based sequencing
methods [9] are being explored in a dual nanopore system for a cost effective,
high throughput, chemical-free, and real time barcode generation.

Figure 1:  Schematics of a model dsDNA captured in cylindrical nanopore of
diameter $d=2\sigma$ and thickness $t_{pore}$, where $\sigma$ is the diameter
of each monomer (purple beads). Protein tags (barcodes) of the same diameter
but of different colors (only three are shown in here) interspersed along the
dsDNA backbone. Opposite but unequal forces $\vec{f}_{U}$ and $\vec{f}_{D}$
are applied to straighten the dsDNA as it translocates in the direction bias
net $\pm|\Delta\vec{f}_{UD}|=\pm|\vec{f}_{U}-\vec{f}_{D}|$ through the
nanopore. (b) Positions of the protein tags along the contour length of the
model dsDNA of length $L=1024\sigma$ which represents an actual dsDNA of 48500
base pairs. The location of the tags are listed in Table-I.

The possibility of determining DNA barcodes have been demonstrated in a dual
nanopore device, by scanning a captured dsDNA multiple times by applying a net
periodic bias across the two pores [9, 10, 11, 12]. Theoretical and simulation
studies have also been reported in the context of a double nanopore system
[13, 14, 15]. In this article, we investigate a similar strategy in silico in
a cylindrical nanopore and demonstrate that a cylindrical nanopore can have a
competitive advantage over a dual nanopore system. By studying a model dsDNA
with barcodes using Brownian dynamics we establish an important result that it
is due to the disparate dwell time and speed of the barcodes (“tags”) compared
to the nucleotide segments

Table 1: Tag positions along the dsDNA

Tag # | $T_{1}$ | $T_{2}$ | $T_{3}$ | $T_{4}$ | $T_{5}$ | $T_{6}$ | $T_{7}$ | $T_{8}$
---|---|---|---|---|---|---|---|---
Position | 154 | 369 | 379 | 399 | 614 | 625 | 696 | 901
Separation | 154 | 215 | 10 | 20 | 215 | 11 | 71 | 205

(“monomers”) the current blockade time information only is not enough and will
lead to an inevitable underestimation of the distance between the barcodes.
Furthermore, using the ideas of the tension propagation theory [16, 17], we
demonstrate that information about the fast-moving nucleotides in between the
barcodes,- not easily accessible experimentally is a key element to resolve
the underestimation. We suggest how to obtain this information experimentally
and provide a physically motivated “two-step” interpolation scheme for an
accurate determination of barcodes, even when the separation of (unknown) tags
has a broad distribution.

$\bullet$ The Model System: Our in silico coarse-grained (CG) model of a dsDNA
consist of 1024 monomers interspersed with 8 barcodes at different locations
shown in Fig. 1 and Table-I is motivated by an experimental study by Zhang et
al. on a 48500 bp long dsDNA with 75 bp long protein tags at random locations
along the chain [10, 11, 12] using a dual nanopore device. Here we explore if
a cylindrical nanopore with applied biases at each end can resolve the
barcodes with similar accuracy or better. We purposely choose positions of the
8 barcodes (Table-I) to study how the effect of disparate distances among the
barcodes affects their measurements. The tags  $T_{2}$,  $T_{3}$,  $T_{4}$ are
closely spaced and form a group. Likewise, another group consisting of
$T_{5}$ and  $T_{6}$ are put in a closer proximity to  $T_{7}$. The tags
$T_{1}$ and  $T_{8}$ are further apart from the rest of the tags. The general
scheme of the BD simulation strategy for a translocating homo-polymer under
alternate bias has been discussed in our recent publication [13, 14] and in
the Appendix A.

In this article, tags are introduced by choosing the mass and friction
coefficient at tag locations to be different than the rest of the monomers
along the chain. This requires modification of the BD algorithm as discussed
in the Appendix A. The protein tags used in the experiments [10, 11, 12]
translate to about three monomers in the simulation. The heavier and extended
tags introduce a larger viscous drag. Instead of explicitly putting side-
chains at the tag locations, we made the mass and the friction coefficient of
the tags 3 times larger. This we find enough to resolve the distance between
the tags. Two forces $\vec{f}_{U}$ and $\vec{f}_{D}$ at each end of the
cylinder in opposite directions keep the DNA straight inside the channel and
allows translocation in the direction of the net bias (please see Fig. 1 and
Fig. 2).

$\bullet$ Barcodes from repeated scanning: As potentially could be done in a
nanopore experiments, we switch the differential bias once the first tag or
the last tag ( $T_{1}$ ,  $T_{8}$) translocates through the nanopore during
up($U$)/down($D$) $\rightarrow D/U$ translocation yet having end segments
inside the pore (please see Fig. 2) so that the DNA remains captured in the
cylindrical pore and the barcodes are scanned multiple times.

Figure 2: Demonstration of the epoch when the bias voltage is flipped. (a)
showing the last barcode is yet to translocate in the downward direction when
the net bias $\Delta\vec{f}_{DU}=\vec{f}_{D}-\vec{f}_{U}>0$. (b) shows the
situation after a later time when finally all the barcodes crossed the
cylindrical pore during downward translocation with a portion of the end
segment still remaining inside the pore. At this point the bias is flipped
with an upward bias $\Delta\vec{f}_{UD}=\vec{f}_{U}-\vec{f}_{D}>0$,
translocation now occurs in the upward direction. In this way, the DNA remains
captured all the time during repeated scans. Figure 3: (a) Demonstration of
calculation of wait time for  $T_{7}$ which has the monomer of index $m=696$.
The dwell velocity is then calculated using Eqn. 2. (b) Demonstration of
calculation of tag time delay $\tau_{78}^{U\rightarrow D}=t_{i}^{U\rightarrow
D}(8)-t_{i}^{U\rightarrow D}(7)$ for tags  $T_{7}$ and  $T_{8}$ while they are
moving downward. Please note that similar quantity for upward translocation
$\tau_{87}^{D\rightarrow U}=t_{i}^{D\rightarrow U}(7)-t_{i}^{U\rightarrow
D}(8)\neq\tau_{78}^{U\rightarrow D}$ as there is no symmetry of the tags along
the chain.

The question we ask: can we recover the actual barcode locations from these
scanning measurements, so that the method can be applied to determine unknown
barcodes ? We monitor two important quantities, - the dwell time of each
monomer and the time delay of arrival of two successive monomers at the pore
as demonstrated in Fig. 3 and explained below. For each up/down-ward scan we
measure the dwell times of the monomer $m$ as follows:

$\displaystyle W^{U\rightarrow D}(m)=t_{f}^{U\rightarrow
D}(m)-t_{i}^{U\rightarrow D}(m),$ (1a) $\displaystyle W^{D\rightarrow
U}(m)=t_{f}^{D\rightarrow U}(m)-t_{i}^{D\rightarrow U}(m).$ (1b)

Here $t_{i}^{U\rightarrow D}(m)$ and $t_{f}^{U\rightarrow D}(m)$ are the
arrival and exit times of the monomer with index $m$ as further demonstrated
in Fig. 3(a). The corresponding dwell velocities $v_{dwell}^{U\rightarrow
D}(m)$ and $v_{dwell}^{D\rightarrow U}(m)$ for the $m^{th}$ bead (either a
monomer or a tag) along the channel axis (please see Fig. 3(a)) can be
obtained as follows.

$\displaystyle v_{dwell}^{U\rightarrow D}(m)=t_{pore}/W^{U\rightarrow D}(m),$
(2a) $\displaystyle v_{dwell}^{D\rightarrow U}(m)=t_{pore}/W^{D\rightarrow
U}(m).$ (2b)

In an actual experiment one measures the dwell velocities of the tags only
which are equivalent to the current blockade times.

$\bullet$ Non uniformity of the dwell velocity: The presence of tags with
heavier mass ($m_{tag}=3m_{bulk}$) and larger solvent friction
($\gamma_{tag}=3\gamma_{bulk}$) introduces a large variation in the dwell time
and hence a large variation in the dwell velocities of the DNA beads and tags
(see Fig. 4). In general, there is no up-down symmetry for the dwell
time/velocity as tags are not located symmetrically along the chain backbone.
Thus the physical quantities are averaged over $U\rightarrow D$ and
$D\rightarrow U$ translocation data. The average dwell velocity
$\bar{v}_{dwell}(m)=\frac{1}{2}\left[v_{dwell}^{U\rightarrow
D}(m)+v_{dwell}^{D\rightarrow U}(m)\right]$ clearly shows two different
velocity envelopes - the tags residing at the lower envelope. Fig. 4 shows
that

Figure 4: Dwell velocity of monomer in a cylindrical nanopore system.
$\triangledown$ and $\vartriangle$ represent downward and upward
translocation. $\circ$ are average of both directions. Filled triangles and
circles correspond to tag dwell velocities.

the dwell velocities of the tags (green circle ) are significantly lower than
the velocity of the nucleotides in between the tags, which will underestimate
the barcode distances as explained later. We further notice that increasing
the pore width resolves the barcodes better.

$\bullet$ Barcode estimation using a cylindrical nanopore setup: If the dsDNA
with barcodes were a rigid rod, then one could obtain the barcode distances
$d_{mn}^{U\rightarrow D}$ and $d_{nm}^{D\rightarrow U}$ between tags $T_{m}$
and $T_{n}$ from the following equations (shown for downward translocation
only):

$\displaystyle d_{mn}^{U\rightarrow D}$ $\displaystyle=v_{mn}^{U\rightarrow
D}\times\tau_{mn}^{U\rightarrow D}\quad{\rm where,}$ (3a) $\displaystyle
v_{mn}^{U\rightarrow D}$
$\displaystyle=\frac{1}{2}\left[v_{dwell}^{U\rightarrow
D}(m)+v_{dwell}^{U\rightarrow D}(n)\right],$ (3b)
$\displaystyle\tau_{mn}^{U\rightarrow D}$
$\displaystyle=\left(t_{i}^{U\rightarrow D}(n)-t_{i}^{U\rightarrow
D}(m)\right).$ (3c)

Here $\tau_{mn}^{U\rightarrow D}$ is the time delay of arrivals of $T_{m}$ and
$T_{n}$ for downward translocation (please see Fig. 3(b) which explains the
special case when $m=7$ and $n=8$). Similar Equations can be obtained by
flipping $D$ and $m$ with $U$ and $n$ respectively. In other words, Eqn. 3
gives the shortest distance and not necessarily the contour length (the actual
distance) between the tags. However, this is the only data accessible through
experiments and likely to provide an underestimation of the barcodes. Fig.
5(a) shows the data for 300 scans. The average with error bars are shown in
the 3rd column of Table-II. Excepting for  $T_{6}$ these measurements grossly
underestimate the actual positions with large error bars.

Figure 5: (a) Barcodes generated using different methods. In each graph, the colored symbols/lines refer to the corresponding colors of the barcodes  $T_{1}$,  $T_{2}$ ,  $T_{3}$, , $T_{4}$,  $T_{5}$,  $T_{6}$,  $T_{7}$, and  $T_{8}$ respectively. The open and filled symbols represent barcodes for $U\rightarrow D$ and $D\rightarrow U$ transolcation using (a) Eqn. 3; (c) using method 1, and (e) using method 2. In (b), (d) and (e) the solid lines refer to the actual location of the barcodes and the dashed lines correspond to the averages from (a), (c) and (e) respectively. The improved accuracy for the latter two methods are readily visible in (d) and (f) where the simulation and the actual data are almost indistinguishable. Table 2: Barcodes from various methods Tag  | Relative  | Barcode  | Barcode  | Barcode
---|---|---|---|---
Label  | Distance  | (Eqn. 3)  | (Method-I)  | (Method-II)
| w.r.t $T_{5}$ | $\times$ | $\checkmark$ |  $\checkmark$
$T_{1}$ | 460  | 373 $\pm$ 122  | 459 $\pm$ 59  | 460 $\pm$ 43
$T_{2}$ | 245  | 197 $\pm$ 67  | 250 $\pm$ 39  | 250 $\pm$ 32
$T_{3}$ | 235  | 183 $\pm$ 63  | 237 $\pm$ 38  | 237 $\pm$ 32
$T_{4}$ | 215  | 167 $\pm$ 54  | 211 $\pm$ 35  | 211 $\pm$ 30
$T_{5}$ | 0  | 0  | 0  | 0
$T_{6}$ | 11  | 11 $\pm$ 3  | 14 $\pm$ 4  | 11 $\pm$ 3
$T_{7}$ | 82  | 68 $\pm$ 23  | 86 $\pm$ 23  | 86 $\pm$ 21
$T_{8}$ | 287  | 230 $\pm$ 73  | 287 $\pm$ 65  | 287 $\pm$ 73

$\bullet$ Tension Propagation (TP) Theory explains the source of discrepancy
and provides solution: Unlike a rigid rod, tension propagation governs the
semi-flexible chain’s motion in the presence of an external bias. In TP theory
and its implementation in Brownian dynamics, the motion of the subchain in the
cis side decouples into two domains [16, 17]. In the vicinity of the pore, the
tension front affects the motion directly while the second domain remains
unperturbed, beyond the reach of the TP front. In our case, after the tag
$T_{m}$ translocates through the pore, preceding monomers are dragged into the
pore quickly by the tension front, analogous to the uncoiling effect of a rope
pulled from one end. The onset of this sudden faster motion continues to grow
and reaches its maximum until the tension front hits the subsequent tag
$T_{m\pm 1}$, with larger inertia and viscous drag. At this time (called the
tension propagation time [18]) the faster motion of the monomers begins to
taper down to the velocity of the tag $T_{m\pm 1}$. This process continues
from one segment to the other. Fig. 6 shows an example on how the segment
connecting $T_{7}$ and  $T8$ has non-monotonic velocity under the influence of
the tension front.

Figure 6:  Tension propagation (TP) through the chain backbone connecting
$T_{7}$ and  $T8$. (a) Figure shows a sudden fast movement of monomers right
after $T_{7}$’s passage through the pore. Due to the TP front’s influence
(yellow blob region), subsequent monomers are sucked into the pore quickly.
(b) TP front finally reaches  $T8$, leading to a slower translocation speed
due to the tag’s large inertia and higher viscous drag.

These contour lengths of faster moving segments in between two barcodes are
not accounted for in Eqn 3. The experimental protocols are limited in
extracting barcode information through Eqn. 3 (measuring current blockade
time) and therefore, likely to underestimate the barcodes, unless the data is
corrected to account for the faster moving monomers in between two tags.

$\bullet$ How to determine the barcodes correctly ? Fig. 1(b) and the $3^{rd}$
column of Table-II when looked closely provide clues to the solution of the
underestimated tag distances. We note that locations of the isolated tags
(such as,  $T_{1}$ and  $T_{8}$) far from  $T_{5}$ have a larger error bar
while  $T_{6}$ which is adjacent to  $T_{5}$ has the correct distance from
Eqn. 3. It is simply because in the later case the contour length between
$T_{5}$ and  $T_{6}$ is almost equal to the shortest distance. Evidently, the
error bars increase with increased separation.

To compare the barcodes obtained from Eqn. 3 with the actual contour length
(see $2^{nd}$ column of Table-II) between tag pairs, we invoke the Flory
theory to determine the scaling exponent $\nu$ [19] which reveals the behavior
of the segments under translocation. The heatmap in Fig. 7 confirms that when
the separation between the tag pairs is less compared to the DNA length, the
connecting segment behaves like a rigid rod ($\nu>0.6$). While for the
isolated tags, $\nu<0.6$ suggests that barcodes are shorter than their
respective contour lengths. This clarifies the reason behind the barcode
underestimation for the tags which are spaced apart while yielding accurate
barcodes for tags located in groups.

Figure 7:  Flory exponent ($\nu$) for the segment connecting a tag pair
represented as a two dimensional heatmap array on the color scale ranging from
blue to white.

Within the experimental set up we suggest the following two methods which will
account for the larger velocities of the monomers.

Method 1 - Barcode from known end-to-end Tag distance: In order to measure the
barcode distances accurately one thus needs the velocity of the entire chain.
If the distance between  $T_{1}$ and  $T_{8}$) $d_{18}\simeq L$, then the
velocity of the segment $d_{18}$ will approximately account for the average
velocity of the entire chain $v_{chain}$ and correct the problem as
demonstrated next. First we estimate the velocity of the chain

$v_{chain}^{U\rightarrow D}\approx v_{18}^{U\rightarrow
D}=d_{18}/\tau_{18}^{U\rightarrow D},$ (4)

assuming we know $d_{18}$ and $\tau_{18}^{U\rightarrow D}$ is the time delay
of arrival at the pore between  $T_{1}$ and  $T_{8}$ for ${U\rightarrow D}$
translocation. We then estimate the barcode distance $d_{mn}^{U\rightarrow D}$
between tags $T_{m}$ and $T_{n}$ as

$d_{mn}^{U\rightarrow D}=v_{18}^{U\rightarrow D}\times\tau_{mn}^{U\rightarrow
D}.$ (5)

In the similar fashion one can calculate $d_{mn}^{D\rightarrow U}$ using
$v_{chain}^{D\rightarrow U}$ and $\tau_{mn}^{U\rightarrow D}$ information
respectively. How do we know $d_{18}$ ? One can use $d_{18}\approx L_{\rm
scan}$ and $v_{chain}\approx\bar{v}_{\rm scan}$, from Eqn. 6 where
$\bar{v}_{\rm scan}$ is the the average velocity of the scanned length $L_{\rm
scan}$ from repeated scanning as discussed in the next paragraph. This method
is effective for estimating the long-spaced barcodes but it overestimates the
barcode distance if multiple barcodes are close by as evident in Fig. 5(d) and
the $4^{th}$ column of table-II. Thus, we know how to obtain barcode distances
accurately when they are close by (from Eqn. 3) and for large separation (Eqn.
5). We now apply the physics behind these two schemes to derive an
interpolation scheme that will work for all separations among the barcodes.

Method 2 - Barcode using two-step method: Average scan time $\bar{\tau}_{\rm
scan}$ for the entire chain (which can be measured experimentally) is a better
way to estimate the average velocity of the chain. $L_{\rm scan}$ is the
maximum length up to which the dsDNA segment remains captured inside the
nanopore gets scanned and denotes the theoretical maximum beyond which the
dsDNA will escape from the nanopore, thus, $L\approx L_{\rm scan}$. For
example, in our simulation, scanning length $L_{\rm scan}=0.804L$. We denote
the average scan velocity as

$\bar{v}_{\rm scan}=\frac{1}{N_{\rm scan}}\sum_{i=1}^{N_{\rm scan}}L_{\rm
scan}/\tau_{\rm scan}(i),$ (6)

where $\tau_{\rm scan}(i)$ is the scan time for the $i^{th}$ event, and
$N_{\rm scan}=300$. To proceed further, we use our established results that
the monomers of the dsDNA segments in between the tags move with velocity
$\bar{v}_{\rm scan}$, while tags move with their respective dwell velocities
$v_{mn}^{U\rightarrow D}$ and $v_{mn}^{D\rightarrow U}$ (Eqn. 2). We then
calculate the segment velocity between two tags by taking the weighted average
of the velocities of tags and DNA segment in between as follows.

First, we estimate the approximate number of monomers
$N_{mn}=d_{mn}^{U\rightarrow D}/\langle b_{l}\rangle$ ($\langle b_{l}\rangle$
is the bond-length) by considering the tag velocities only using Eqn. 3. We
then calculate the segment velocity accurately by incorporating weighted
velocity contributions from both the tags and the monomers between the tags.

$\begin{split}v_{weight}^{U\rightarrow
D}=\frac{1}{N_{mn}}\Big{[}v_{dwell}^{U\rightarrow
D}(m)&+v_{dwell}^{U\rightarrow D}(n)+\\\ &(N_{mn}-2)\bar{v}_{\rm
scan}\Big{]}\end{split}$ (7)

The barcodes are finally estimated by multiplying the calculated 2-step
velocity in Eqn. 7 above by the tag time delay as

$d_{mn}^{U\rightarrow D}=v_{weight}^{U\rightarrow
D}\times\tau_{mn}^{U\rightarrow D}$ (8)

for $U\rightarrow D$ translocation and repeating the procedure for
$D\rightarrow U$ translocation. This 2-step method accurately captures the
distance between the barcodes when the two tags are in proximity or spaced
apart from each other. Table-II and Fig. 5 summarize our main results and
claims.

$\bullet$ Summary & Future work: Motivated by the recent experiments we have
designed barcode determination experiment in silico in a cylindrical nanopore
using the Brownian dynamics scheme on a model dsDNA with known locations of
the barcodes. We have carefully chosen the locations of the barcodes so that
the separations among the barcodes span a broad distribution. We discover that
if we use the dwell time data only for the barcodes from multiple scans of the
dsDNA to calculate the average velocities of the tags then the method
underscores the barcode distances for tags further apart. Our simulation
guides us to conclude that the source of this underestimation lies in
neglecting the information contained in the faster moving DNA segments in
between any two tags. We use non-equilibrium tension propagation theory to
explain the non-monotonic velocity of the chain segments where the barcodes
lie at the lower bound of the velocity envelope as shown in Fig. 4. The
emerging picture readily shows the way how to rectify this error by
introducing an interpolation scheme that works well to determine barcodes
spaced apart for all distances which we validate using simulation data. We
suggest how to implement the scheme in an experimental setup. It is important
to note that the interpolation scheme-based concept of the TP theory is quite
general and we have ample evidence that this will work in a double nanopore
system as well.

$\bullet$ Conflicts of interest: The authors declare no competing financial
interest.

$\bullet$ Acknowledgements: The research at UCF has been supported by the
grant number 1R21HG011236-01 from the National Human Genome Research Institute
at the National Institute of Health. All computations were carried out at the
UCF’s high performance computing platform STOKES.

##  Appendix A The Model and Brownian dynamics simulation

Our BD scheme is implemented on a bead-spring model of a polymer with the
monomers interacting via an excluded volume (EV), a Finite Extension Nonlinear
Elastic (FENE) spring potential, and a bond-bending potential enabling
variation of the chain persistence length $\ell_{p}$ (Fig.A1). The model,
originally introduced for a fully flexible chain by Grest and Kremer [20], has
been studied quite extensively by many groups using both Monte Carlo (MC) and
various molecular dynamics (MD) methods [21]. Recently we have generalized the
model for a semi-flexible chain and studied both equilibrium and dynamic
properties [18, 22, 23] and studied compression dynamics of a model dsDNA
inside a nanochannel [24, 25] . The mutual EV interaction among any two
monomers are given by the truncated Lennard-Jones (LJ) potential with a cut-
off radius $2^{1/6}\sigma$

$\displaystyle
U_{LJ}(r_{ij})=\begin{cases}4\epsilon\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right]+\epsilon,\text{
for }r<2^{1/6}\\\ 0,\text{ otherwise }\end{cases}$ (9)

where $\sigma$ is the effective diameter of a monomer and $\epsilon$ is the
interaction strength. To mimic the connectivity between two adjacent monomers,
finite-extensible-non-linear elastic (FENE) potential

Figure A1: (a) Illustration depicts the monomers are interacting via LJ and
FENE potential. The three body bending potential is calculated using the angle
$\theta_{i}$ between two adjacent bond vectors $\vec{b}_{i}$ and
$\vec{b}_{i+1}$ respectively. (b) Interaction potential between two
consecutive monomers is given by the green line for a separation distance $r$
in unit of $\sigma$. The blue diamonds denote the LJ potential with a cutoff
radius $2^{1/6}\sigma$ and the magenta circles correspond to the FENE
potential with a spring constant $\kappa_{F}=30.0\epsilon/\sigma^{2}$. (c) A
cylindrical nanopore of diameter $2\sigma$ is dilled into a material of
thickness $t_{pore}$. The walls consist of purely repulsive LJ particles.

$\displaystyle
U_{FENE}(r_{ij})=-\frac{1}{2}\kappa_{F}R_{0}^{2}\ln\left[1-\left(\frac{r_{ij}}{R_{0}}\right)^{2}\right]$
(10)

is used with the maximum bond-stretching length $R_{0}=1.5\sigma$ and spring
constant $\kappa_{F}=30\epsilon/\sigma^{2}$. Here,
$r_{ij}=|\vec{r}_{i}-\vec{r}_{j}|$ is the separation distance between two
adjacent monomers $i$ and $j=i\pm 1$ located at $\vec{r}_{i}$ and
$\vec{r}_{j}$ respectively. Along with these two potentials, we introduce a
bending potential

$\displaystyle
U_{bend}(\theta_{i})=\kappa\left(1-\cos\left(\theta_{i}\right)\right)$ (11)

with bending rigidity $\kappa$. In three dimensions, for $\kappa\neq 0$, the
persistence length $\ell_{p}$ of the chain is related to $\kappa$ via [26]

$\ell_{p}=\frac{\kappa}{k_{B}T},$ (12)

where $k_{B}$ is the Boltzmann constant and $T$ is the temperature. Here
$\theta_{i}$ is the bond angle between two subsequent bond vectors
$\vec{b}_{i}=\vec{r}_{i+1}-\vec{r}_{i}$ and
$\vec{b}_{i-1}=\vec{r}_{i}-\vec{r}_{i-1}$. A cylindrical nanopore of diameter
$2\sigma$ is drilled through a solid material of thickness $t_{pore}$ consists
of immobile and purely repulsive LJ particles. Our model of DNA polymer
consists $1016$ monomer beads along with $8$ heavier tags ($T_{1}$ \- $T_{8}$)
located at positions $154,369,379,399,614,625,696$, and $901$ respectively
(please refer to Fig. 2 and Table-I in the main article). A recent study by
Zhang et al. on 48512 bp long dsDNA uses 75 bp long protein tags as barcodes
[10]. In simulation, we purposely choose the mass of a tag ($m_{tag}$) three
times heavier of a normal monomer to replicate the tags used in the
experiments. We proportionally increase the solvent friction of the tags
$\Gamma_{tag}=3\Gamma_{i}$. We use the Brownian dynamics to solve the equation
of motion of a monomer $i$ having a mass $m_{i}$ and solvent friction
$\Gamma_{i}$ as

$\displaystyle
m_{i}\ddot{\vec{r_{i}}}=\vec{\nabla_{i}}\left[U_{LJ}+U_{FENE}+U_{bend}+U_{wall}\right]-\Gamma_{i}\vec{v_{i}}+\eta_{i}$
(13)

where $\Gamma_{i}=0.7\sqrt{m_{i}\epsilon^{2}/\sigma^{2}}$ is the frictional
coefficient arising from solvent-monomer interaction. For the case of a tag,
$m_{tag}=3m_{i}$ and $\Gamma_{tag}=2.1\sqrt{m_{i}\epsilon^{2}/\sigma^{2}}$.
The Gaussian white noise $\eta_{i}$ arising from thermal fluctuation is delta
correlated and expressed as
$\langle\eta_{i}(t).\eta_{j}{j}(t^{\prime})\rangle=2dk_{B}T\Gamma\delta_{ij}\delta(t-t^{\prime})$
with $d=3$ in three dimension. We express length and energy in units of
$\sigma$ and $\epsilon$ respectively such that $k_{B}T/\epsilon=1.0$. The
parameters for FENE potential in Eq. (10) are $\kappa_{F}$ and $R_{0}$, and
set to be $\kappa_{F}=30\epsilon/\sigma^{2}$ and $R_{0}=1.5\sigma$. The
numerical integration of Eq. (13) is implemented using the algorithm
introduced by Gunsteren and Berendsen [27]. Our previous experiences with BD
simulation suggests that for a time step $\Delta t=0.01$ these parameters
values produce stable trajectories over a very long period of time and do not
lead to unphysical crossing of a bond by a monomer [22, 23]. The average bond
length stabilizes to $\langle b_{l}\rangle=0.971\pm 0.001\sigma$ with
negligible fluctuation regardless of the chain size and rigidity [22]. Hence
we relate the polymer’s contour length $L$ and the number of monomers $N$ as
$L=(N-1)\langle b_{l}\rangle$.

## References

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