# Field-extension statistics of charged semiflexible polymers stretched with
uniform electric fields

Ananya Mondal<EMAIL_ADDRESS>Center for Theoretical Biological Physics, Rice
University, Houston, Texas 77005, USA  Greg Morrison  Center for Theoretical
Biological Physics, Rice University, Houston, Texas 77005, USA  Department of
Physics, University of Houston, Houston, Texas 77204, USA

###### Abstract

Single-molecule force-extension experiments have enabled the manipulation of
biomolecules in unprecedented detail. These studies have allowed quantitative
measurements of the mechanical responses of biomolecules to applied forces
explaining their roles in key biological functions. Electrophoretic stretching
of charged polymers such as DNA in uniform electric fields is one such
example, currently, used for sequencing purposes. Field-extension statistics
of charged polymers differ from laser optical tweezer setups for force-
extension experiments due to a non-uniform tension along the backbone of the
chain, the effects of which remain poorly understood. In this paper, we modify
an existing analytically tractable mean-field (MF) approach to account for the
heterogeneity in tension for electric fields. This model has been shown to
successfully predict the force-extension relations of inextensible polymers
under uniform tension. Naively using this model for stretching of charged
polymers such as DNA under electric fields results in local overstretching of
the chain and gives inaccurate field-extension statistics. We improve this
approach and account for the inhomogeneity in the tension by subdividing the
chain into smaller segments while imposing the inextensibility of the chain.
We find that the subdivided MF model shows better agreement with the
simulations for the force-extension plots, and can predict the extension of
the chain at any applied field value. We also show that using an isotropic
mean-field model overestimates the longitudinal fluctuations both for tension
forces as well as for fields. We correct the quantitative predictions for the
longitudinal fluctuations in the mean extension by numerically differentiating
the field-extension plots. We also find that the subdivided MF model can
accurately predict the statistics of experimentally relevant quantities, such
as transverse fluctuations, due to the analytical tractability of the model.
These field-extension predictions may be further used to introduce confinement
effects in the subdivided MF model and develop a comprehensive understanding
of sequencing technologies.

##  I Introduction

###  I.1 Stretching of biomolecules

Single-molecule force-extension experiments Bustamante et al. (2003); Smith et
al. (1992) have permitted the manipulation of biomolecules, and studying their
responses to external fields has provided useful insights into the behavior of
these biologically relevant molecules. These experiments are often conducted
in a constant force ensemble Brower-Toland et al. (2002); Cui and Bustamante
(2000); Kulić and Schiessel (2004) (e.g. laser optical tweezers), and the
force-extension relations are well described with a wormlike chain model
Rubinstein et al. (2003) under uniform tension Ha and Thirumalai (1995); Marko
and Siggia (1995). DNA is a polyelectrolyte (PE), and in many contexts, it is
more convenient to stretch charged bio-molecules by applying a constant
electric field. Electrophoresis Stellwagen (2009) has been a popular
separation technique for sorting DNA based on its molecular size. This
technique has also been shown Gurrieri et al. (1999); Long et al. (1996) to be
useful for observation purposes of long DNA polymers in agarose, by pinning
the molecule and elongating it using large electric fields.

More recently, technical advancements have made it possible to create
nanochannels Yeh and Szeto (2016); Mahshid et al. (2018); Roy et al. (2017);
Dorfman et al. (2013) and nanopores Belkin et al. (2015); Shi et al. (2017)
through which polyelectrolytes tethered at one end can be elongated under
electric fields. The ability of confined channels to suppress thermal
fluctuations Chen (2017); Suma et al. (2018); Morrison (2020) and observe
linear DNA has made electrophoretic stretching an active area of research with
growing applications in nanopore-DNA sequencing Belkin et al. (2015); Heerema
and Dekker (2016) and consequent identification of epigenetic modifications
Cortini et al. (2016). Electric field-driven stretching of DNA inside
nanopores is currently used in low-cost long-read genome sequencing devices.
One of the major advantages of using the nanopore sequencing technique
compared to stretching charged polymers like DNA through biological pores is
the prevention of interactions with the pore surface. In this sequencing
method, one end of the DNA molecule is pinned with optical tweezers and the
other end is subject to a constant electric field through a nanopore.
Confinement effects Morrison and Thirumalai (2009); Chen (2016) can also give
rise to complex statistics like back folding or hairpin formation, or
diffusion of knots. This technique holds promise in making gene sequencing
affordable for individual patient diagnostics but a problem lies in the
precision of nucleotide sequence reading is still an open problem Heng et al.
(2005); Qiu et al. (2015, 2021); Lu et al. (2011); Wanunu (2012). To improve
the precision it is important to understand the physical properties of the
charged polymer while it is stretched under a field through molecular
simulations and a mathematical model.

Stretching of single biomolecules with a constant tension is well-studied in
laser optical tweezer experiments Bustamante et al. (2003) but unlike these
systems, stretching with an electric field generates a non-uniform tension
profile along the backbone as seen in experiments Gurrieri et al. (1999); Roy
et al. (2017); Marko and Siggia (1995); Wanunu (2012). Electric fields Liedel
et al. (2012, 2013) and shear flows Majewski and Yager (2015) induced
alignment in artificial block co-polymers are also being used to study the
effects of local stress fields on different morphology and improve
nanofabrication techniques. A common feature reported in electric field
stretching experiments Klotz et al. (2017); Roy et al. (2017) is inhomogeneous
tension created along the backbone of the polyelectrolyte. The tension
gradient increases from zero at the free end to a maximum at the tethered end
of the polymer. The inhomogeneity in the tension depends upon the charge
distribution on the charged polymer but even for a uniformly charged polymer,
the chain develops a non-uniform tension profile. The charge distribution,
however, impacts the screening effects from the condensation of counter-ions
Manning (1969) that results in an effective charge distribution on the chain.
The effective screening of DNA is also influenced by the viscous drag from the
surrounding solvent molecules. This effect is called an electro-osmotic flow
Ghosal et al. (2019); Luan and Aksimentiev (2008), which is in the opposite
direction to the field stretching. Since the hydrodynamic drag is competing
with the electric field stretching so the strength of this force determines
how much the chain is linearly stretched and the strength is determined by the
confinement radius of the nanochannels.

To utilize the full potential of electrophoretic stretching and sequencing
technology, it is important to shed light on the statistical properties of
polymers under inhomogeneous tension. In this paper, we study a simpler case
of charged polymers stretched in an electric field. As a stepping stone to
more higher-order interactions, we devise an analytically tractable solution
for predicting field-extension plots for a single semiflexible filament pinned
at one end and stretched with an electric field. The mean-field model used in
this paper describes the semiflexible biomolecule using the persistence length
of the chain that can be described as an effective electrostatic persistence
length Skolnick and Fixman (1977) based on the screened charge density on the
chain. Hydrodynamic and confinement effects however are future avenues for
this project.

###  I.2 Mathematical models

Many biomolecules and their assemblies are often described using a wormlike
chain (WLC) Rubinstein et al. (2003), or the related discrete Kratky Porod
Model Kratky and Porod (1949); Doi et al. (1988), which account for the
inextensibility and intrinsic resistance to bending of these polymers. The
competition between entropy (which prefers the formation of a random coil) and
the energetic cost of bending (which prefers a straight conformation) leads to
statistics in the WLC that differ significantly from flexible polymer
modelsRubinstein et al. (2003). The persistence length $l_{p}$ is a measure of
the energetic cost required to bend the polymer away from a straight
configuration. Despite the WLC model being physically reasonable, it is often
formidable to extract equilibrium or dynamical properties analytically Hori et
al. (2007); Samuel and Sinha (2002). In order to develop analytically
tractable techniques to study the statistics of a WLC, a number of useful
approximate theories have been developed over the years. Theoretical studies
Marko and Siggia (1995); Ha and Thirumalai (1997); Bucci and Spakowitz (2020);
Hori et al. (2007); Jacobson et al. (2017); Benetatos and Frey (2004); Bleha
and Cifra (2019); Everaers et al. (2021) have used equilibrium statistical
mechanics to model experimental observable such as the mean end-to-end
extension of WLCs when pulled under tension. For strongly stretched WLCs under
large forces, the chain only slightly fluctuates in the direction transverse
to the applied force. This behavior is approximated as the weakly bending rod
(WBR) limitBenetatos and Terentjev (2010); Marko and Siggia (1995); Andersen
et al. (2021). Using the WBR limit, Marko and Siggia (MS) Marko and Siggia
(1995) have derived an interpolated force-extension equation from low to high
values of a uniform tension force, $f$, that remains widely used. At low
forces, the chain extends linearly, much like a Hookean spring $\langle
R_{ee}\rangle/L\propto f$, where $\langle R_{ee}\rangle$ is the mean end-to-
end extension of the chain in the direction of the applied force $f$). For
higher forces, where WLCs such as DNA are stretched between $30-95\%$ of its
contour length, the difference between the mean end-to-end extension and the
total contour length goes to zero as $1/f^{1/2}$ Marko and Siggia (1995),
which is also accounted for in the MS interpolation. The deviation of a WLC
from a freely jointed chain’s extension is accurately captured by the
interpolation equation in Marko and Siggia (1995).

MS Marko and Siggia (1995) and others Andersen et al. (2021); Petrosyan
(2017); Bouchiat et al. (1999), have recognized that it is a harder problem to
solve the interpolation equation for inhomogeneous forces such as a constant
electric field. A reasonable approximation to make meaningful predictions for
force extension due to an electric field uses the WBR limit as most of the
chain is strongly stretched and oriented in the direction of the applied
field. From this, one obtains a scaling for the strongly-stretched regime,
where the difference between the mean end-to-end extension and the total
contour length also goes to zero as $1/\epsilon^{1/2}$, with $\epsilon$ being
a dimensionless electric field Marko and Siggia (1995). Due to the similarity
in scaling with tension, MS argued that the electric field can be treated as
an effective uniform tension in the WBR limit. The main challenge is to then
predict the force-extension statistics in the intermediate region when thermal
fluctuation is comparable to electrostatic stretching energy. In this
intermediate region, it would be useful to have a simple interpolation
(similar to MS) or an analytical expression that predicts the extension of the
polymer. Although an exact field-extension curve can be derived, the
calculation is quite laborious Hori et al. (2007); Bleha and Cifra (2019), and
involves an infinite series of Airy functions. One of the aims here is to find
a simpler formula for experimentalists to use to analyze their data.

In this paper, we use an existing analytically tractable mean-field theory Ha
and Thirumalai (1995, 1997); Morrison and Thirumalai (2009) that has
successfully predicted the force-extension statistics of a semiflexible
polymer stretched with uniform tension forces and improve on the model to
predict a force-extension formula for electric fields. Lagowski et al Lagowski
et al. (1991), proposed a functional integral formalism that uses a mean-field
approach to yield analytically tractable results for wormlike chains without
using the WBR limit. Winkler et al Winkler et al. (1994) have demonstrated a
procedure that calculates the partition function for discrete stiff chains
exactly, using the maximum entropy principle. The mean-field model was further
adopted by Ha and Thirumalai Ha and Thirumalai (1995, 1997) and applied to the
problem of semiflexible polymers stretched using uniform tension. The
mathematical convenience of the mean-field method has proved fruitful in
reproducing the scaling given by Marko and Siggia Marko and Siggia (1995) and
also in creating a recipe to examine other useful quantities such as
fluctuations and correlation functions. Furthermore, the mean-field method has
been successful in producing tractable theories for biologically relevant
systems such as kinetics of loop formation in WLCs Hyeon and Thirumalai (2006)
and confinement of biopolymers Morrison and Thirumalai (2009); Morrison
(2020). Benetatos and Frey Benetatos and Frey (2004) have made an interesting
contribution to calculating the linear response of a grafted polyelectrolyte
to a uniform electric field using spherical harmonics for different limiting
conditions. Hori et al Hori et al. (2007) have studied the stretching of short
biopolymers using electric fields in the WBR limit and have shown finite-size
effects have a significant role to play in the field-extension statistics.
Inspired by these theoretical studies, in this paper, our objective is to
construct a minimally formidable mean-field technique that not only provides a
pathway to derive field-extension statistics of WLCs in electric fields but
also comes up with a workable formula for experimentalists to test our theory.

This paper begins with an introduction to a mean-field model Ha and Thirumalai
(1995); Morrison and Thirumalai (2009) used for predicting the force-extension
statistics for worm-like chains stretched with uniform forces. In Section II.1
we describe the mathematical framework of the MF model in the context of WLCs
stretched in electric fields, and we demonstrate that the MF model does not
account for the inhomogeneity in the tension for the case of the electric
field stretching. We show the underlying homogeneity assumption in the
stretching parameter in the MF model leads to overstretching of the bonds and
hence fails to replicate the experimental behavior of WLCs in a high electric
field stretching. To improve the MF model we propose a ‘subdivided mean-field
(MF) theory’ in Section II.3, dividing the chain into subdivisions
corresponding to variable stretching parameters. The simulation methodology is
described in Section III. In Section IV we predict the wide range of
statistics, including field-extension plots, longitudinal fluctuations, and
transverse fluctuations using the ‘subdivided MF model’. We find that
simulations agree better with the field-extension statistics proposed here
because in this model, the inhomogeneous tension is correctly accounted for.
In Section IV.4, we show that the longitudinal fluctuations predicted from the
MF approach are overestimated due to the isotropy in MF model. We also compare
the FEC plots with the widely used interpolation formula proposed by Marko et
al. Marko and Siggia (1995), where the underlying assumption is an effective
force can replace the electric field. In Section IV.2, we show that the
assumption of the Marko-Siggia (MS) equation is not accounted for intermediate
values of the electric field where the inhomogeneity in the tension is most
pronounced. Finally, we suggest a new method to improve the prediction of
longitudinal fluctuations in Section IV.2. We conclude with a summary of our
results, applicability, and limitations.

##  II Methods

###  II.1 Homogeneous mean-field (MF) model for WLCs stretched in a uniform
electric field

Figure 1: Coarse-grained bead spring model for worm-like chains (WLCs) – The
coarse-grained bead spring model for worm-like chains has two energetic
contributions, namely, bending energy represented by the persistence length
($l_{p}$) and stretching energy between the bonds represented by the parameter
$\lambda$.

The Gaussian bead-spring model Doi et al. (1988) is the simplest mathematical
representation for long flexible polymers. Many biomolecules are semiflexible
having an internal stiffness that needs to be accounted for in the model. The
worm-like chain (WLC) model has been used to describe the statistics of
biomolecules that are semiflexible like DNA, and actin. An inextensible
wormlike chain, parameterized by the arc length $s$ and with position
$\mathbf{r}(s)$ has the Hamiltonian
$H_{wlc}=\frac{l_{p}}{2}\int_{0}^{L}ds(\partial_{s}\hat{\mathbf{\mathbf{u}}})^{2}$,
with
$\hat{\mathbf{\mathbf{u}}}=\partial_{s}\mathbf{\mathbf{r}}/|\partial_{s}\mathbf{\mathbf{r}}|$
the local unit tangent vector and $\partial_{s}$ a derivative with respect to
$s$. The WLC Hamiltonian represents the resistance to bending, with stiffer
chains having longer persistence lengths $l_{p}$ (where $l_{p}=\kappa/k_{B}T$
where $\kappa$ is the bending modulus of the chain and $T$ the temperature).
Statistical averages using this Hamiltonian must account for the constraint of
fixed length $L$, and numerous papers Hori et al. (2007); Lagowski et al.
(1991) have determined the statistics of WLCs in various potentials (usually
expressed as a series expansion that cannot easily be reduced to elementary
functions). In many cases, relaxing the rigid constraint of inextensibility
permits analytically tractable predictions in terms of elementary functions,
such as the Weakly Bending Rod (WBR) and Mean Field (MF) approaches. In the MF
approach, the rigid bond constraints are replaced by an average length and are
introduced in the Hamiltonian as the inextensibility constraint using Lagrange
multipliers. This assumption of averaging out the bond lengths is referred to
as a Mean-Field (MF) model and it has successfully predicted experimentally
relevant quantities in several studies Ha and Thirumalai (1995); Morrison and
Thirumalai (2009); Winkler (2003); Morrison (2020); Mondal and Morrison (2022)
involving biopolymers.

For a charged semiflexible chain of uniform linear charge density $\sigma$,
and length $L$, placed in an electric field
$\boldsymbol{E}=E\boldsymbol{\hat{x}}$, the electrostatic energy is
$-\sigma\beta\mathbf{E}\cdot\int_{0}^{L}ds\
\mathbf{r}(s)=-\sigma\beta\mathbf{E}\cdot\int_{0}^{L}ds\int_{0}^{s}ds^{\prime}\mathbf{u}(s^{\prime})$.
In this section, we use the MF model that works well for uniform tension and
use it naïvely for the electric field. We find that an underlying homogeneity
assumption in the stretching resistances of the bonds leads to the failure of
the MF model in this case. The homogeneity assumption in the MF model causes
the bonds near the tether to overstretch compared to the bonds near the free
end because of a non-uniform tension profile due to the electric field. The
stretching resistances do not account for inhomogeneous stretching and so we
hypothesize the homogeneous MF model fails to capture the correct statistics
in this case. We improve upon this model in section II.3 but before we do that
let us take a closer look into the homogeneous MF model Hamiltonian. The MF
Hamiltonian for a charged WLC in an electric field is

$\displaystyle\beta
H_{0}[\mathbf{u}(s)]=\delta_{0}\mathbf{u}_{0}^{2}+\delta_{L}\mathbf{u}_{L}^{2}+\frac{l_{0}}{2}\int_{0}^{L}ds\dot{\mathbf{u}}^{2}+\lambda\int_{0}^{L}ds\mathbf{u}^{2}-\boldsymbol{\epsilon}\cdot\int_{0}^{L}ds\int_{0}^{s}ds^{\prime}\mathbf{u}(s^{\prime})\;,$
(1)

where $l_{0}$ is a ‘mean-field persistence length’ ($l_{0}\neq l_{p}$),
$\lambda$ is the Lagrange multiplier to introduce the mean-field
inextensibility constraint as resistance to stretching,
$\mathbf{u}=\dot{\mathbf{r}}$ is the local stretching of the chain in the
continuum limit and $\epsilon=\sigma\beta E$ is the dimensionless electric
field force. The chain is pinned at $\mathbf{r}(0)$ end to avoid a tumbling
Smith et al. (1999) or formation of knots Tang et al. (2011). For a positively
charged chain ($\sigma>0$) the charges align in the direction of the electric
field, stretching out the chain linearly. In the absence of an electric field,
the terms involving $\delta$ are required to recover the known bending
correlation function Morrison (2020) using the MF approach, suppressing excess
endpoint fluctuations. Pinning of one end causes an asymmetry in the
fluctuations at each endpoint and so, the Lagrange multipliers or resistance
to stretching parameters at the endpoints are respectively,
$\lambda_{0}=\lambda+\delta_{0}$ at the pinned end and
$\lambda_{L}=\lambda+\delta_{L}$. The MF approach chooses the parameters
$\lambda$, $\delta_{0}$ and $\delta_{L}$, such that the global inextensibility
condition on the length is imposed on average, with
$\langle\int_{0}^{L}ds\mathbf{u}^{2}\rangle_{0}=L$,
$\langle\mathbf{u}^{2}_{0}\rangle_{0}=1$, and
$\langle\mathbf{u}^{2}_{L}\rangle_{0}=1$ (with $\langle...\rangle_{0}$ is the
statistical ensemble average using the MF Hamiltonian from Equation 1). The
mean-field persistence length $l_{0}$ for any freely fluctuating WLC and its
relationship with the true persistence length $l_{p}$ can be determined using
exactly known bond correlations Ha and Thirumalai (1995, 1997); Morrison
(2020). For three-dimensional, the relation is $l_{0}=3l_{p}/2$ but for two-
dimension, the relation is more complicated.

Unlike stretching by a uniform tension as is generally calculated in the
literature Ha and Thirumalai (1995), stretching a WLC in uniform electric
field results in a non-uniform tension profile as also reported in experiments
Gurrieri et al. (1999); Roy et al. (2017); Stellwagen and Stellwagen (2009);
Keyser et al. (2006); Maier et al. (2002). The inhomogeneity in tension along
the backbone originates from the electrostatic energy term,
$U_{\epsilon}[\mathbf{r}]=-\boldsymbol{\epsilon}\cdot\int_{0}^{L}ds\
\mathbf{r}(s)$. For a positively charged chain, the dimensionless energy at an
arc length of $s_{1}<s_{2}$ is $-\boldsymbol{\epsilon}\cdot\int_{0}^{s_{1}}ds\
\mathbf{r}(s)>-\boldsymbol{\epsilon}\cdot\int_{0}^{s_{2}}ds\ \mathbf{r}(s)$ is
true on average, but not for all configurations like for a low electric field
value. It is energetically more favorable for the chain to be aligned with the
field on the tethered end as this ensures aligning a higher density of
positive charges with the direction of the field. The electric field term can
be further simplified by reversing the order of the double integrals and this
results in the term shortening to
$-\boldsymbol{\epsilon}\cdot\int_{0}^{L}ds(L-s)\mathbf{u}$, with
$\mathbf{u}=\partial_{s}\mathbf{r}(s)$. The inhomogeneity in the tension is
also visible from the $\sigma(L-s)$ term because again for $s_{1}<s_{2}$, the
total positive charges that are not aligned with the field are greater for
$\sigma(L-s_{1})$ than for $\sigma(L-s_{2})$. This is equivalent to the same
argument that there is a non-uniform tension profile along the backbone giving
rise to a more aligned chain near the tether and the other end is
comparatively more freely fluctuating. In contrast, for optical tweezers
experiments, the pulling force applied is generally uniform, which necessarily
means each point on the polymer chain feels the same amount of pull.

Combining $U_{\epsilon}[\mathbf{r}]$ with Equation 1, produces a quadratic
Hamiltonian that is straightforward to work with. We complete the squares
using the transformation,
$\mathbf{v}(s)=(v_{x},v_{y},v_{z})=(u_{x}-\epsilon(L-s)/2\lambda,u_{y},u_{z})$
and
$\dot{\mathbf{v}}(s)=(\dot{u_{x}}+\epsilon/2\lambda,\dot{u_{y}},\dot{u_{z}})$.
The transformed MF Hamiltonian is given by

$\displaystyle\beta H_{0}[\mathbf{v}(s)]=\delta_{0}$
$\displaystyle\left(v_{x}(0)+\frac{\epsilon
L}{2\lambda}\right)^{2}+\delta_{0}\left(v_{y}(0)^{2}+v_{z}(0)^{2}\right)+\delta_{L}\mathbf{v}_{L}^{2}+\frac{l_{0}}{2}\int_{0}^{L}ds\dot{\mathbf{v}}^{2}+\lambda\int_{0}^{L}ds\mathbf{v}^{2}$
$\displaystyle-\frac{l_{0}\epsilon}{2\lambda}(v_{x}(L)-v_{x}(0))-\int_{0}^{L}ds\
\frac{\epsilon^{2}(L-s)^{2}}{4\lambda}+\int_{0}^{L}ds\
\frac{l_{0}\epsilon^{2}}{8\lambda^{2}}\;.$ (2)

The quadratic Hamiltonian makes free energy calculation convenient for a WLC,
$\mathcal{F}_{0}=-\log\left[\int\mathcal{D}[\mathbf{v}(s)]e^{-\beta
H_{0}}\right]-\lambda L-\delta_{0}-\delta_{L}$, with the Lagrange multipliers
given by free energy minimization conditions,
$\partial\mathcal{F}/\partial\lambda=\partial\mathcal{F}/\partial\delta_{0}=\partial\mathcal{F}/\partial\delta_{L}=0$.
The free energy calculation reduces to the well-known classical path integral
for a harmonic oscillator Feynman (2018). The free energy is a $d$-dimensional
Gaussian integral of the form $-\log\left[\int\mathcal{D}[\mathbf{v}(s)]\
\exp(-\
\mathbf{V}^{T}\mathbf{A}\mathbf{V})\exp(\mathbf{B}^{T}\cdot\mathbf{V}+\mathbf{C})\right]$,
where $\mathbf{A}$ is a tri-diagonal coefficient matrix of the quadratic
terms, $\mathbf{B}$ is the coefficient matrix of the linear terms,
$\mathbf{C}$ is a constant term matrix (a detailed path integral calculation
in the Supplemental Information), and
$\mathbf{V}=(\mathbf{v}_{0},\mathbf{v}_{L})^{T}$ is the column matrix for the
endpoint variables.

For zero electric field, the scaling is the same as any force-free WLC Ha and
Thirumalai (1995) and the mean-field conditions give, $\sqrt{\lambda
l_{0}/2}=\delta=3/4$. For the intermediate electric field values, the scaling
is only determined numerically as this region has a mix of power laws. In the
high electric field region, the scaling for the Lagrange multipliers with the
electric field is obtained from the mean-field conditions, $\lambda\sim
L\epsilon/2\sqrt{3}$, $\delta_{0}\sim 0.341\sqrt{Ll_{p}\epsilon}$, and
$\delta_{L}\sim-0.465\sqrt{Ll_{p}\epsilon}$, where $l_{p}$ is the true
persistence length of the chain. The scaling of the Lagrange multipliers with
the electric field is obtained from the numerical solutions of the mean-field
equations. The difference in the constraining conditions for the two ends of
the chain is reflected in the Lagrange multipliers $\delta_{0}$ and
$\delta_{L}$, which, for sufficiently high fields, is either positive or
negative. A positive Lagrange multiplier or stretching parameter imparts
harder resistance to stretch compared to a negative value.

###  II.2 Overstretching due to homogeneity assumption in the MF model

We expect that the homogeneous MF model does not account for the inhomogeneity
in tension for WLCs stretched in an electric field and leads to over-
stretching and compression of the bonds. It is energetically favorable for
positively charged monomers to move in the direction of the field, causing the
bonds to align with the electric field. While aligning with the electric
field, the bonds do not stretch equally as they would have done if the polymer
was pulled using a uniform tension Marko and Siggia (1995); Ha and Thirumalai
(1997). For a positively and uniformly charged polymer stretched in an
electric field directed away from the tethered end as shown in Figure 3, the
bonds closer to the tethered end stretch more compared to the bonds closer to
the free end. The bonds near the tether prefer to stretch and not compress
because, in the latter scenario, a larger number of the positive charges will
be pulled out of the electric field making it an energetically unfavorable
configuration. We define overstretching of the bonds as the extension of the
local bond lengths beyond the fixed lengths while maintaining the global
inextensibility of the chain. The overstretching effect of the electric field
gives rise to an inhomogeneous stretching or tension-pulling profile. This
inhomogeneous tension along the backbone of the chain is not a characteristic
of the tether but is retained even in the bulk of long chains. The MF model
assumes that the stretching parameters are equal for all bonds irrespective of
their location in the electric field meaning that the bonds closer to the
tether experience greater stretching forces locally, and can over-stretch for
intermediate or high values of electric field. Similarly, the bonds closer to
the fluctuating end experience less stretch and hence are effectively
compressed.

Figure 2: Homogeneity assumption in the mean-field model leads to
overstretching of bonds – Bond correlation ratio,
$\chi(s)=\langle\int_{0}^{s}ds^{{}^{\prime}}u_{x}^{2}(s^{\prime})\rangle/\langle\int_{s}^{L}ds^{{}^{\prime}}u_{x}^{2}(s^{\prime})\rangle$
plotted against the dimensionless field $\epsilon Ll_{p}$ for 3 different
values of the arc lengths, s = L/4 (blue), s = L/2 (red), s = 3L/4 (black).
Points indicate numerical solutions for the quantity $\chi(s)$ obtained from
the $n=0$ mean-field model in Equation 1. Overstretching of the bonds is a
result of the mean-field assumption of homogeneous stretching parameters,
$\lambda(s)=\lambda$, in the case of electric fields. For all 3 cases, the
ratio picks up as the field is increased, which indicates that the bonds
closer to the tethered end (s = 0) are stretched more compared to the other
half of the chain closer to the free end (s= L). The plots show that the ratio
of the overstretching and compression of the bonds is greater than 1 as $s$ is
increased from $L/4$ to $3L/4$. This means that the overstretching is not
compensated by the compression of the bonds or in other words, the chain does
not attain full extension even for high electric fields.

To quantify the overstretching or compression in bonds as a function of the
applied electric field, we evaluate the bond correlation ratio,
$\chi(s)=\langle\int_{0}^{s}ds^{{}^{\prime}}u_{x}^{2}(s^{\prime})\rangle/\langle\int_{s}^{L}ds^{{}^{\prime}}u_{x}^{2}(s^{\prime})\rangle$
for different values of the arc length $s$ on the chain. The quantity,
$\chi(s)$, is also calculated using the auxiliary field method (detailed in
section IV.1), where
$-\alpha_{1}\int_{0}^{s}ds^{{}^{\prime}}u_{x}^{2}(s^{\prime})$ and
$-\alpha_{2}\int_{s}^{L}ds^{{}^{\prime}}u_{x}^{2}(s^{\prime})$ terms are
introduced to the undivided chain Hamiltonian in Equation 1 as before. Taking
a derivative of the free energy with respect to the corresponding auxiliary
field,
$\partial(-\mathcal{F}^{\alpha})/\partial\alpha_{i}\Big{|}_{\alpha_{i}=0}$,
yields the ratio. This ratio is plotted for different values of $s$ in Figure
2. The ratio of all the curves for different arc lengths is initially low for
low fields and then picks up as the field is increased and finally saturates
because of the contour length inextensibility constraint. A low ratio means
that the bonds are equally stretched as expected for low fields. For higher
values of the ratio, the bonds nearer to the tethered end are stretched more
compared to the ones beyond the arc length $s$ or closer to the free end at
$s=L$. The plots are shown for three different values of $s=L/4,L/2,3L/4$ to
observe the effect of overstretching and compression due to the underlying
homogeneity assumption in the undivided MF model.  For $s=L/4$, $\chi<1$ for
$\epsilon\rightarrow 0$ and reaches $\chi\approx 1$ for
$\epsilon\rightarrow\infty$ indicating overstretching and compression of the
bonds on either side of $s=L/4$ of a polymer in an applied field. The $s=L/2$
(red) and $s=3L/4$ (black) plots show that the overstretching and compression
ratio of the bonds is much greater than one for high electric field values and
even for low and intermediate ranges of the field. This shows that the
homogeneous MF model without subdivisions cannot give the correct statistics
for the field-extension curves.

###  II.3 Subdivided chain mean-field model for WLCs stretched in a uniform
electric field

The end-to-end extension for WLC is often used as an observable in force-
extension experiment Bustamante et al. (2003) and plotted with the stretching
force to determine the mechanical properties of WLCs. For a charged WLC
stretched in an electric field, our aim is also to quantitatively predict the
mean-extension and fluctuation statistics of the WLC as a function of the
applied field. The MF model homogeneity assumption in the stretching
parameter, $\lambda(s)=\lambda$, causes all the bonds to resist stretching
equally, which means it fails to balance the inhomogeneity in the backbone
tension due to the electric field. Bonds near the pinned end overstretch and
bonds closer to the free end compress for large electric fields with $\lambda$
as a constant stretching resistance. The homogeneity assumption works well for
uniform tension Ha and Thirumalai (1995) but fails to account for the
inhomogeneity in tension for electric fields. The inhomogeneity is not exactly
accounted for in the homogeneous MF model, and the chain does not stretch to
its full extension, $L$, even for a high field because the over-extension does
not compensate for the compression for non-uniform stretching forces.

Figure 3: Schematic diagram of the subdivided mean-field (MF) theoretical
model accounting for the inhomogeneity in the tension due to electric field
stretching – (A) A positively charged worm-like chain of length $L$ is
tethered at one end (blue bead) and is freely fluctuating on the other. A
uniform electric field, $\mathbf{E}=E\boldsymbol{\hat{x}}$ stretches the chain
as the positive charges align in the direction of the field. The chain is
stretched more on the tethered end than the free end because of inhomogeneity
in tension. To account for the inhomogeneity in tension, we divide the chain
into subdivisions, where each subdivision has a different stretching
resistance. The length of each subdivision becomes vanishingly small for a
large number of subdivisions. (B) The variable stretching parameters for 4
subdivisions are shown in $n=4$ different colors, where
$\lambda_{1}>\lambda_{2}>\lambda_{3}>\lambda_{4}$ for the $n=4$ subdivisions.
The increasing order of the parameters is determined as a result of stationary
phase conditions.

To improve on the homogeneous MF model, we propose a subdivided MF model as
shown in Figure 3. The length of each subdivision is $L/n$, where $n$ is the
total subdivisions. Dividing the chain means that the MF assumption on the
bonds is now locally applied over each subdivision. Physically, this leads to
an over- or under-stretch of the subdivisions but by making the regions
smaller and smaller the effect of over or under-stretching is expected to be
mitigated. Every region now over- or under-stretches, but the net effect is
that the inextensibility is globally satisfied (even if not locally). The mean
field assumption for the $j^{th}$ subdivision is given by
$\langle\int_{j}ds_{j}\mathbf{u}_{j}^{2}\rangle=L/n$, where
$\int_{j}\equiv\int_{(j-1)L/n}^{jL/n}$ and $\mathbf{u}_{j}$ is the stretching
in the $j$th subdivision. The subdivided chain MF Hamiltonian from Equation 1
becomes

$\displaystyle\beta
H_{0}[\mathbf{u}_{j}(s)]=\delta_{0}\mathbf{u}_{0}^{2}+\delta_{L}\mathbf{u}_{L}^{2}+\frac{l_{0}}{2}\int_{0}^{L}ds\dot{\mathbf{u}}^{2}+\sum_{j=1}^{n}\lambda_{j}\int_{j}ds_{j}\mathbf{u}_{j}^{2}-\boldsymbol{\epsilon}\cdot\int_{0}^{L}ds\int_{0}^{s}ds^{\prime}\mathbf{u}(s^{\prime})\;,$
(3)

where $\int_{j}$ is the integral over the segment $s=(j-1)L/n$ to $s=jL/n$ and
$\lambda_{j}$ is the stretching parameters for the $j^{th}$ subdivision.

Completing the squares on the Hamiltonian in Equation 3, with the following
transformation for the subdivisions,
$\mathbf{v}_{j}=(v_{xj},v_{yj},v_{zj})=(u_{xj}-(L-s)\epsilon/2\lambda_{j},u_{yj},u_{zj})$,
results in the new Hamiltonian

$\displaystyle\beta H_{0}[\mathbf{v}_{j}(s)]=$
$\displaystyle\sum_{j=1}^{n}\left(\frac{l_{0}}{2}\int_{j}ds_{j}\dot{\mathbf{v}}_{j}^{2}+\lambda_{j}\int_{j}ds_{j}\mathbf{v}_{j}^{2}-\int_{j}ds_{j}\
\frac{\epsilon^{2}(L-s_{j})^{2}}{4\lambda_{j}}+\int_{j}ds_{j}\
\frac{l_{0}\epsilon^{2}}{8\lambda_{j}^{2}}\right)$
$\displaystyle-\sum_{j=1}^{n}\left(\frac{l_{0}\epsilon}{2\lambda_{j}}v_{x}(s=jL/n)+v_{x}(s=(j-1)L/n)\right)$
$\displaystyle+\delta_{0}\left(v_{x}(0)+\frac{\epsilon
L}{2\lambda_{1}}\right)^{2}+\delta_{0}\left(v_{y}(0)^{2}+v_{z}(0)^{2}\right)+\delta_{L}\mathbf{v}_{L}^{2}\;.$
(4)

The free energy of the subdivided MF Hamiltonian in Equation 4 is calculated
using Gaussian integrals,
$\mathcal{F}_{n}=-\log\left[\int\mathcal{D}[\mathbf{v}(s)]e^{-\beta
H_{0}[\mathbf{v}_{j}(s)]}\right]-\sum_{j=1}^{n}\lambda_{j}L/n-\delta_{0}-\delta_{L}$.
The path integral calculation is detailed in Supplemental Information. The
distribution of the Lagrange multipliers follows from the mean-field
equations,
$\partial\mathcal{F}_{n}/\partial\lambda_{j}=\partial\mathcal{F}_{n}/\partial\delta_{0}=\partial\mathcal{F}_{n}/\partial\delta_{L}=0$.
From the mean-field equations we obtain that in the zero electric field limit,
all the stretching parameters are equal,
$\lambda_{j}=9/8l_{0},\delta_{0}=3/4,\delta_{L}=3/4$, which is expected for a
WLC Ha and Thirumalai (1995).

Figure 4: Variation of stretching parameters derived from the subdivided mean-
field model as a function of position on the WLC –Numerical solutions (solid
dots) for $\lambda(s)$ as a function of arc length $s$ is plotted for
different values of the fields. The lines are fit using the equation
$\lambda(s)\sim(L-s)\epsilon/2$. For very low field (blue) the $\lambda$
values are constant for all subdivisions and are close to the zero field value
of $\lambda_{0}=9/8l_{0}=0.05$ for $l_{p}=15$. As the field is increased the
variation in the stretching parameters for $n=64$ subdivisions is observed.
The variation is nonlinear for intermediate values of the field (shown in the
inset). For high values of the field, the $\lambda(s)$ has a clear linear
dependence on the arc length, and the stretching parameters closer to $s=0$ is
higher compared to the ones closer to $s=L$. This order of the stretching
parameters is a result of the inhomogeneous stretching tension on either side
of the chain. A higher value of stretching resistance ensures that the bonds
are harder to overstretch and a lower value of $\lambda(s)$ leads to less
compression leading to correct extension statistics.

Figure 4 shows the variation of the stretching parameters for $n=64$
subdivisions and different field values. The points are the numerical
solutions, and solid lines are fits. For a low field, all the roots are the
same and have a value very close to the values for zero fields,
$\lambda=9/8l_{0}$. For intermediate field values, the inset plots show the
variation of the stretching parameters more clearly, as in this range of the
field, the scaling laws are not very clear. With an increase in the field, the
stretching parameters closer to $s=0$, or the pinned end, are higher than for
the subdivisions closer to the free end. It means that the bonds towards the
pinned end are harder to stretch, and the variable distribution of
$\lambda_{j}$ corrects the issue of over-stretching and compression of bonds.
For a high value of the field, the trend of decreasing $\lambda_{j}$ remains,
and it fits the scaling law $\lambda(s)\sim(L-s)\epsilon/2$ perfectly. We
hypothesize that the issue of over-extension and compression of the bonds is
corrected by using this distribution of stretching parameters, which makes it
harder to overstretch bonds on the pinned end of the chain or compress bonds
closer to the free end of the chain. We predict that accounting for the
inhomogeneity in tension using the subdivided MF method results in full
extension of the WLC in high electric fields.

###  II.4  Are the partition functions equivalent before and after the
completion of squares?

To obtain a quadratic Hamiltonian we complete squares in Equations 2 and 4.
The partition functions calculated from the Hamiltonian before and after the
completion of squares should be equal. To demonstrate the equivalence of the
partition function before and after the completion of squares, the propagators
in the partition functions (see Supplemental Information for details on
propagators) must be equal as well. For example, in the subdivided chain, let
us consider the propagators connecting the $j^{th}$ and $(j+1)^{th}$
subdivisions. For the adjacent propagators of $j^{th}$ and $(j+1)^{th}$
segments to be connected at the joining point,
$v_{xj}=u_{xj}-(L\epsilon/2\lambda_{j})(1-j/n)$ from the $j^{th}$ segment and
$v_{xj}=u_{xj}-(L\epsilon/2\lambda_{j+1})(1-j/n)$ from the $(j+1)^{th}$
segment, should be the same. The only difference between the connection point
is the $\lambda_{j}$ or $\lambda_{j+1}$ parameter depending on if the $v_{xj}$
point is in the $j^{th}$ segment propagator or $(j+1)^{th}$ segment
propagator. The connecting point, $v_{xj}$ in the $(j+1)^{th}$ propagator can
also be rewritten as,
$v_{xj}=u_{xj}-(L\epsilon/2)(1-j/n)(\frac{1}{\lambda_{j}}+\eta_{j})$, where
$\eta_{j}=(1/\lambda_{j+1}-1/\lambda_{j})$ is the difference in the inverse of
the stretching parameters for the adjacent subdivisions. For a large number of
subdivisions ($n\rightarrow\infty$), the difference $\eta_{j}\approx 0$ as the
stretching parameters scale as $1/\lambda_{j}\sim 1/n$. It follows that for a
large number of subdivisions, the connecting point $v_{xj}$ is the same for
adjacent propagators and hence the equivalence of the partition functions
holds true before and after the completion of squares. This equivalence is
trivial for the non-subdivided chain.

##  III Simulations

In order to quantify the predictive power of the subdivided chain model, we
performed MCMC simulations to measure the mean and fluctuations of various
observables. We simulate a coarse-grained chain of $N=250$ monomers and the
length of the chain is given by $L=(N-1)a$, where $a$ is the distance between
monomers. The chain is generated by growing $(N-1)$ bond vectors,
$\hat{\mathbf{u}_{i}}$, each with a fixed length. There are two energetic
contributions: the bending energy from the intrinsic stiffness (persistence
length $l_{p}$) of a worm-like chain and from the electric field. In our
simulations, the bending energy is
$U_{\text{bend}}=\kappa\sum_{i}(1-\hat{\mathbf{u}}_{i}\cdot\hat{\mathbf{u}}_{i+1})$
with $\kappa$ as the bending stiffness parameter related to the persistence
length,
$l_{p}/a\approx(\kappa-1+\kappa\coth\kappa)/2(\kappa+1-\kappa\coth\kappa)$
Dorfman et al. (2013), where $a$ is the distance between the beads. We used
three different values of the persistence length in the simulations,
$l_{p}=2,15,30$ monomers. In our simulations, we take a positively and
homogeneously charged chain with a charge density $\sigma$. The electric field
energy is introduced in our simulations as a dimensionless field,
$\epsilon=\sigma\beta E$, and varies from low values of $\epsilon=10^{-5}$ to
moderate and high values of $\epsilon=1$, where the polymer stretches to its
full contour length. The electric field energy in the simulation is $\Delta
U_{E}=-\epsilon(N-i)$dx, $i$ is a randomly picked bond to be displaced to
generate a trial configuration in the Monte Carlo simulations and dx is the
change in the x-component of the bond for the trial configuration which is
being aligned with the field.

Initially, each chain is grown randomly (without regard for the electric
field). At each Monte Carlo step a random bond $i$ is chosen uniformly along
the chain, and replaced with a new trial at an angle $\theta_{i}$ with respect
to the existing $(i-1)^{th}$ bond. $\theta_{i}$ is drawn from
$P(\theta_{i})\propto e^{-a\cos(\theta_{i})/l_{p}}$ (with no dependence on the
electric field), and is accepted or rejected with the Metropolis criterion
$p_{acc}=\mbox{max}(1,e^{-\beta\Delta U})$, with $U=U_{bend}+\Delta U_{E}$.
Using this algorithm we perform approximately $10^{8}$ MC steps between the
initially grown chain and the final configuration. After equilibration, a
total of $10000$ chains are generated to obtain field-extension statistics
such as end-to-end distance or fluctuations at each combination of stiffness
and applied field parameters.

##  IV Results

###  IV.1 Field-extension (FEC) plots

Force-extension experiments measure the end-to-end extension of single
molecules Bustamante et al. (2003), and can help determine the mechanical
properties of biomolecules like persistence length. We calculate the mean end-
to-end extension of the chain aligned with the direction of the field for low,
intermediate, and high values of the applied electric field. We expect that
the undivided MF model in Section II.1 does not predict the correct field-
extension statistics due to overstretching/compression due to the
inhomogeneous tension. The mean end-to-end extension in the direction of the
field ($\mathbf{E}=E\boldsymbol{\hat{x}}$) is the sum of all the bond vectors
aligned with the field, $\langle
X_{ee}\rangle=\left\langle\int_{0}^{L}dsu_{x}(s)\right\rangle$. We calculate
this quantity by adding an auxiliary field term,
$-\alpha\int_{0}^{L}dsu_{x}(s)$ (see details in Supplemental Information), to
the Hamiltonian in Equation 1. Completing the squares for the new Hamiltonian
using the transformation, $w_{x}=u_{x}-((L-s)\epsilon+\alpha)/2\lambda$,
enables us to calculate the average end-to-end extension analytically,
$\langle
X_{ee}\rangle=-\frac{\partial\mathcal{F}^{\alpha}}{\partial\alpha}\Big{|}_{\alpha=0}$.
For a long chain, the bulk of the chain contributes largely compared to the
endpoints to the mean end-to-end extension, and in this limit, $\langle
X_{ee}\rangle=\frac{L^{2}\epsilon}{4\lambda}$. The stretching parameter,
$\lambda$, is determined from the mean-field equation,
$\frac{\partial\mathcal{F}^{\alpha}}{\partial\lambda}\Big{|}_{\alpha=0}=0$ and
we neglect $\delta_{0}$ and $\delta_{L}$ as endpoint effects should not
dominate the global extension and we neglect $\delta_{0}$ and $\delta_{L}$ as
endpoint effects should not dominate the global extension. Simultaneously
solving the mean-field equation for the stretching parameter, $\lambda$ and
the mean end-to-end extension expression in the long chain limit, gives the
field-extension equation for the undivided chain to order $1/L$,

$\displaystyle-L+\frac{4X_{ee}^{2}}{3L}+3\sqrt{\frac{X_{ee}}{2l_{0}\epsilon}}=0.$
(5)

Equation 5 has four solutions: two imaginary roots and one real root that
results in a large value of $\langle X_{ee}\rangle$ exceeding the contour
length $L$ (so these roots are excluded), and one real root that converges as
$\epsilon\rightarrow 0$. In the high electric field limit, the mean end-to-end
extension is $\langle X_{ee}\rangle=\sqrt{\frac{3}{4}}L$ for a $\lambda\sim
L\epsilon/2\sqrt{3}$. The end-to-end extension is shorter than the full
contour length for high electric fields due to local compression and
overextension on the mean-field level. This is consistent with the result in
Section II.2.

The mean end-to-end extension for the $n$ subdivisions can also be calculated
by introducing $n$ auxiliary field terms
$-\alpha_{j}\int_{j}ds_{j}u_{x}(s_{j})$ in the subdivided MF Hamiltonian with
total extension, $\langle
X_{ee}\rangle=-\sum_{j=1}^{n}\frac{\partial\mathcal{F}_{n}^{\alpha}}{\partial\alpha_{j}}\Big{|}_{\alpha_{j}=0}$.
From the calculation outlined in Supplemental Information, the free energy can
be calculated and the mean end-to-end extension equation for the $j^{th}$
subdivision is obtained as $\langle
X_{ee}^{j}\rangle=\frac{L^{2}\epsilon(1-2j+2n)}{4n^{2}\lambda_{j}}$. In the
$L\rightarrow\infty$ limit the mean-field equations for the stretching
parameters to order $1/L$ are, $-\frac{L}{n}+\int_{j}ds_{j}\
\frac{(L-s_{j})^{2}\epsilon^{2}}{4\lambda_{j}^{2}}+\frac{3L}{2n\sqrt{2l_{0}\lambda_{j}}}=0$.
To determine the field-extension equations, we can readily solve for
$\lambda_{j}=\frac{L^{2}\epsilon(1-2j+2n)}{4n^{2}\langle X_{ee}^{j}\rangle}$
and this results in the field-extension equations for the $j^{th}$
subdivision,

$\displaystyle-\frac{L}{n}+\frac{4n(1+3j^{2}+3n(1+n)-3j(1+2n))(X_{ee}^{j})^{2}}{3L(1-2j+2n)^{2}}+3\sqrt{\frac{X_{ee}^{j}}{(2-4j+4n)l_{0}\epsilon}}=0.$
(6)

Exact expressions for $X_{ee}^{j}$ are unwieldy, but can be readily evaluated
numerically.

In the limit of $L/n\rightarrow 0$, the total end-to-end extension of the
chain becomes $\langle X_{ee}\rangle=\int_{0}^{L}ds\
\frac{(L-s)\epsilon}{2\lambda(s)}$. The high field behavior of full extension
is recovered with $\lambda(s)\sim(L-s)\epsilon/2$ leading to a full extension
$\langle X_{ee}\rangle/L=1$. Note that the expression, $\langle
X_{ee}\rangle/L=\frac{L\epsilon}{4\lambda}$, is very similar to the uniform
stretching force extension derived by the authors in Ref. Ha and Thirumalai
(1995), where $\langle X_{ee}\rangle/L=\frac{fL}{2\lambda}$ except the
coefficient of $\frac{1}{4}$ instead of $\frac{1}{2}$. The difference in
coefficients is because of the inhomogeneous tension term in the electric
field, $(L-s)$, as compared to the uniform stretching tension force $f$. The
behavior of $\langle X_{ee}\rangle$ with respect to the field strength must be
evaluated numerically as discussed in Section II.3.

Figure 5: Field-extension curves for WLCs – Normalized mean end-to-end
extension $\langle X_{ee}\rangle/L$ of a worm-like chain stretched in a
uniform electric field in the x-direction, is plotted as a function of the
dimensionless electric field $\epsilon Ll_{p}$, where $\epsilon=\sigma\beta
E$. For low fields, the extension grows linearly but for intermediate fields,
the extension is nonlinear due to the bending cost of worm-like chains. At
very high values of the field, the chain stretches to its full contour length
and the extension should saturate at 1 (dashed blue line). For $n=0$
subdivisions (the homogeneous MF model without subdivisions), the high field
limit is wrong as the chain only stretches to $0.86L$. The saturation value
improves as the number of subdivisions is increased. Simulation data shows
good agreement with the numerical solution for the $n=64$ curve (black solid
line) for intermediate fields (even for small $n$). Simulations are shown for
three different values of the persistence length, $l_{p}=2$ (black dots), $15$
(blue dots), $30$ (brown dots) monomers, and $L=250$ monomers. Error bars are
on the order of the symbol sizes.

Figure 5 shows the field-extension curves for different numbers of
subdivisions plotted against the dimensionless electric field, $\epsilon
Ll_{p}$. As calculated using the MF approach for an undivided chain, the full
end-to-end extension in the direction of the field is less than the total
contour length, for high electric fields. Subdivision of the chain that
accounts for the inhomogeneity in the tension for electric fields, improves
the scaling of the extension for high fields. For $n=64$ the total extension
for high fields, $\langle X_{ee}/L\rangle\approx 1$, and in principle, for an
infinite number of subdivisions the chain attains full extension as expected.

###  IV.2 A comparative study with literature

Marko et al. Marko and Siggia (1995) proposed an interpolation equation for
force-extension experimental data of worm-like chains and has been widely used
since. The force-extension equation that fits optical tweezers experiments
data very well is called the MS (Marko-Siggia) equation:

$\displaystyle\beta fl_{p}=\frac{\langle
X_{ee}\rangle}{L}+\frac{1}{4(1-\langle X_{ee}\rangle/L)^{2}}-\frac{1}{4}\;,$
(7)

where $\beta fl_{p}$ is the dimensionless tension force and $\langle
X_{ee}\rangle$ is the extension. MS also derives an analog of the MS equation
for electric fields by equating the tension force with an effective field, and
for strong stretching fields report the relationship between a field and an
‘effective tension’ ($f_{\text{eff}}$) as, $f_{\text{eff}}=\sigma LE/4$, where
$\sigma$ is the charge density of the polymer and $l_{p}$ is the true
persistence length. This approximation is appropriate only in specific limits,
as MS noted in the paper Marko and Siggia (1995). For example, for finite-
length chains in weak fields, the MS approximations are not guaranteed to
produce the correct statistics. This was directly addressed in the MS paper
Marko and Siggia (1995) but is important to recognize when discussing the
effective force felt by a polymer in a moderate field.

MS equation is derived under the assumption of the Weakly Bending Rod limit
which implies that the transverse fluctuations are slight about the force or
field stretching axis. At high fields, the leading order correction to
extension is derived from this assumption, $X_{ee}\approx 1-\frac{1}{2}(\beta
fl_{p})^{-1/2}$. This scaling immediately implies $\beta
f_{\text{eff}}=L\epsilon/4$ in the limit of high field, where $f_{\text{eff}}$
is the effective tension. In the opposite field limit, we expect that even if
the field is weak the effective tension on the entire chain is not negligible.
For low fields, the extension is very close to zero and grows linearly,
$X_{ee}/L\approx\beta fl_{p}$, leading to $\beta
f_{\text{eff}}=\frac{L\epsilon}{2}$. While it appears an effective tension can
be defined for any value of $\epsilon$, the scaling coefficient $\beta
f_{\text{eff}}=c(\epsilon)L\epsilon$ is not constant with $\epsilon$.  The
effective force as a function of the field $\epsilon$ is determined by
plugging in the analytical expression for the mean end-to-end extension,
$\langle X_{ee}\rangle$, obtained from the subdivided MF model equations 6,
into the MS equation (Equation 7). The numerical solutions for the effective
force as a function of the field are shown in Supplemental Information Figure
S2

Figure 6 compares the FECs for the subdivided MF model with the MS model for
two different values of the effective force: namely $\beta
f_{\text{eff}}=L\epsilon/4$ (high fields) and $L\epsilon/2$ (low fields). The
low-field region agrees better with the effective force derived for low fields
using the MS equation and similarly, the high-field FEC agrees better with the
high-field effective force graph. However, none of the MS curves
quantitatively captures the statistics for all three regions. Simulations
agree better with the subdivided chain model in the intermediate region and
although the difference may seem small in Figure 6, due to the normalization.
For long chains, the difference can be significant, especially for long reads
in nanopore sequencing. The discrepancy between simulations and the MS
predictions is greatest in the intermediate region. The underlying assumption
of the MS model that a field can be replaced by an effective tension force is
only valid in certain regions of the field and cannot be generalized with a
common coefficient $c(\epsilon)$.

Figure 6: Comparison of the FECs for WLCs in a field with the
literature–Force-extension plots for a worm-like chain stretched under an
electric field compared to the Marko-Siggia (MS) equation (Equation 7). The
solid black line is the subdivided MF model theory for $n=64$ subdivisions.
The Blue dashed line is the MS equation for the effective force, $\beta
f_{\text{eff}}=\frac{L\epsilon}{4}$ and the red dashed line is the MS equation
for the effective force, $\beta f_{\text{eff}}=\frac{L\epsilon}{2}$. The blue
line agrees more with the subdivided MF theory for high fields, whereas the
red line agrees better with low fields. Simulations are represented by solid
dots with error bars for 3 different values of the stiffness parameters
$l_{p}=2,15,30$. The simulations agree better with the subdivided MF theory
for the intermediate values because this model accounts for the inhomogeneity
and does not approximate the field as an effective tension force.

###  IV.3 Transverse fluctuations

Imaging of stiff biomolecules in an electric field has been captured
experimentally Maier et al. (2002), and it has been observed that the
fluctuations transverse to the applied field tend to be low at the tethered
endpoint and significantly greater at the free end. For low fields, these
fluctuations are larger at every point along the chain, but the fluctuations
at all points along the backbone decrease as the field strength increases (as
the chain becomes more elongated and aligned with the field). To calculate the
transverse fluctuations at $s=kL/n$ on the backbone we add an auxiliary field
term, namely
$-\alpha\int_{0}^{s}ds^{{}^{\prime}}\mathbf{u}^{\perp}(s^{\prime})$ to the
Hamiltonian in Equation 3, where $\mathbf{u}^{\perp}(s^{\prime})$ are the
transverse components in $y$ and $z$ directions. It should be noted here that
since $s$ is an internal point at which the transverse fluctuations are
measured, the path integral is performed using the coordinates $\mathbf{u}$
and not the transformed one,
$\mathbf{v}^{\perp}=\mathbf{u}^{\perp}-\alpha/2\lambda_{i}$ (see details in
Supplemental Information). The Hamiltonian for calculating the transverse
fluctuations after adding the auxiliary term is

$\displaystyle\beta
H_{0}[\mathbf{u}^{\perp}_{j}(s)]=\delta_{0}(\mathbf{u}_{0}^{\perp})^{2}+\delta_{L}(\mathbf{u}_{L}^{\perp})^{2}+\frac{l_{0}}{2}\int_{0}^{L}ds^{{}^{\prime}}(\dot{\mathbf{u}}^{\perp})^{2}$
$\displaystyle+\sum_{j=1}^{n}\lambda_{j}\int_{j}ds^{{}^{\prime}}_{j}(\mathbf{u}_{j}^{\perp})^{2}$
$\displaystyle-\alpha\int_{0}^{s}ds^{{}^{\prime}}\mathbf{u}^{\perp}(s^{{}^{\prime}})\;.$
(8)

The transverse fluctuations
$\langle{\delta\mathbf{r}^{\perp}(s)}^{2}\rangle=\left\langle\int_{0}^{s}ds^{{}^{\prime\prime}}\int_{0}^{s}ds^{{}^{\prime}}\mathbf{u}^{\perp}(s^{{}^{\prime\prime}})\cdot\mathbf{u}^{\perp}(s^{{}^{\prime}})\right\rangle-\left\langle\int_{0}^{s}ds^{{}^{\prime}}\mathbf{u}^{\perp}(s^{{}^{\prime}})\right\rangle^{2}$
are obtained by taking the double derivative of the free energy,
$\partial^{2}(-\mathcal{F}^{\perp})/\partial\alpha^{2}\Big{|}_{\alpha=0}$.

Figure 7: Transverse fluctuations in WLCs stretched using an electric field –
(A) Normalized transverse fluctuations,
$\langle{\delta\mathbf{r}^{\perp}(s)}^{2}\rangle/Ll_{p}$ are plotted against
the arc length ($s/L$). Solid dots represent subdivided MF theory for $n=64$
subdivisions and lines represent simulation data. Simulations were performed
for $L=250$ monomers and $l_{p}=15$ monomers. As the field increases, the
transverse fluctuations decrease. For a lower value of the field (blue) the
fluctuations increase along the backbone of the chain showcasing the effect of
inhomogeneous stretching tension. (B) Snapshots of a trumpet (stem-flower)
configurations show that the chains are aligned with the field more on the
tethered end compared to the free end, which exhibits higher transverse
fluctuations. The effect is more prominent for low fields (blue,
$\epsilon=0.05$).

The subdivided MF model can predict the transverse fluctuations in the long
chain limit as shown in Figure 7. Figure 7 shows the transverse fluctuations
plotted as a function of arc length, $s$, for two different values of the
applied electric field. The fluctuations are small for a high field, where the
chain is fully stretched, but for a moderate value of the electric field, the
fluctuations have a prominent nonlinear increasing profile with the arc
length. The trumpet shape of the transverse fluctuations shown in the
snapshots indicates that the chain fluctuates less on the pinned end than on
the free end due to the inhomogeneous tension in the intermediate and high
fields. For very low fields, the chain is hardly stretched and the transverse
fluctuations are close to the fluctuations in the mean end-to-end distance
squared for a worm-like chain without any force or field. The deviation near
the $s=0$ or pinned end is due to the dropping of the $\delta$ Lagrange
multiplier terms for ease of calculation.

###  IV.4 Longitudinal fluctuations

Fluctuations in the end-to-end distance also vary with the magnitude of the
applied electric field. We calculate the longitudinal fluctuations for the
subdivided chain model as $\langle\delta
X_{ee}^{2}\rangle=\frac{\partial^{2}(-\mathcal{F}_{n}^{\alpha})}{\partial\alpha^{2}}\Big{|}_{\alpha=0}$.
For a WLC, the fluctuations are well known Rubinstein et al. (2003),
$\left\langle\int_{0}^{L}ds\int_{0}^{L}ds^{\prime}u_{x}(s)u_{x}(s^{\prime})\right\rangle=\frac{2}{3}l_{p}L-\frac{2}{3}l_{p}^{2}(1-e^{-L/l_{p}})$.
Our subdivided model should recover these fluctuations for WLCs in the zero
field limit. In the zero-field limit, the chain’s fluctuations in the mean
extension are also the largest because the chain is not stretched at all
compared to intermediate values when the chain is partially stretched or high
field values when the chain is fully stretched, and fluctuations are zero. We
follow the same routine as the mean end-to-end extension in Equation 6 to
solve for the longitudinal fluctuations in a subdivided chain. To determine
the fluctuations for a subdivision indexed $j$, by solving the mean-field
equations for $\\{\lambda_{j}\\}$ and the equations from the double derivative
of the energy with respect to the field $\alpha$,
$\lambda_{j}=\frac{L}{2n\langle\delta X_{ee}^{2}\rangle^{j}}$, where
$\langle\delta X_{ee}^{2}\rangle^{j}$ is the fluctuations for the subdivision
indexed, $j$. The resulting longitudinal fluctuation equations for each
subdivision are

$\displaystyle-\frac{L}{n}+\frac{\langle\delta
X_{ee}^{2}\rangle^{j}L\epsilon^{2}(1+3j^{2}+3n+3n^{2}-3j(1+2n))}{3n}+\frac{3}{2}\sqrt{\frac{\langle\delta
X_{ee}^{2}\rangle^{j}L}{nl_{0}}}=0\;,$ (9)

where $\langle\delta X_{ee}^{2}\rangle^{j}$ are the transverse fluctuations.
Adding the contributions from the fluctuations of all the subdivisions gives
the analytical expression for the total MF longitudinal fluctuations (plotted
in Figure 8 as the solid black line). In Figure 8, the dashed line indicates
the finite size effects from the endpoint terms. The difference between these
two limits is only for the low electric field region when the field is too low
to align the chain and stretch.

Figure 8: Longitudinal fluctuations in the mean end-to-end extension for worm-
like chains stretched in an electric field – Subdivided MF model yields
normalized longitudinal fluctuations ($\langle\delta
X_{ee}^{2}\rangle/Ll_{p}$) for a worm-like chain stretched under an electric
field for $n=64$ subdivisions (black solid line). Inset shows the longitudinal
fluctuations for the case of a uniform tension $f$. Dots with error bars show
simulations for three different values of stiffness parameter: $l_{p}=2$
(black), $l_{p}=15$ (blue), $l_{p}=30$ (brown). The mean-field model does not
predict the longitudinal fluctuations correctly in the case of both the
electric field and the tension (inset) due to the underlying assumption of
isotropy in all directions. Finite size effects (dashed lines) are more
prominent for stiff chains in low electric fields due to contributions from
the exponential term in the absence of a field, with $\langle\delta
X_{ee}^{2}\rangle=\frac{2}{3}l_{p}L-\frac{2}{3}l_{p}^{2}(1-e^{-L/l_{p}})$.
Simulations for $\kappa=2$ and $\kappa=15$ agree with the low electric field.

In Figure 8, fluctuations from the subdivided chain theory are plotted for a
large number of subdivisions with $n=64$. The Monte Carlo simulations agree
well with the fluctuations at low field values. For the intermediate field
values, the fluctuations are over-estimated by the MF model, and for the high
field values, the simulations match our model and the longitudinal
fluctuations are close to zero. To understand if this discrepancy is an effect
of the inhomogeneity of the tension or the MF assumption, we computed the
longitudinal fluctuations of a WLC pulled with a uniform tension force
$\mathbf{f}$. The discrepancy in the intermediate values of the tension force
is also seen for the longitudinal fluctuations. Hinczewski et al. Hinczewski
and Netz (2011) also report the same problem: although the mean end-to-end
extension is correctly predicted by the mean-field model, the underlying
assumption of MF theory is that it is isotropic and does not accurately
differentiate between the longitudinal and transverse fluctuations. The
isotropic mean-field theory, therefore, overestimates the longitudinal
fluctuations, whereas the chain fluctuates in the transverse direction more,
and averaging over all three directions with equal contributions is not
appropriate, ${\delta\mathbf{r}^{\perp}}^{2}\neq\delta X_{ee}^{2}$, where
$\mathbf{r}^{\perp}$ is the position components in the $y$ and $z$ directions.
In the case of uniform tension, the plots for the three different values of
the persistence lengths ($l_{p}$) are shown in the inset of Figure 8.

The failure of the MF model to accurately predict longitudinal fluctuations
originates from the inaccuracy of the isotropic assumption that underlies the
MF model. With no distinction between the stretching stiffness in the
transverse and longitudinal directions, the theory cannot simultaneously
predict the fluctuations in both directions in an analytically exact way. In
order to quantitatively predict the longitudinal fluctuations of the chain, we
exploit the good agreement between simulated and predicted end-to-end
distance. To compute the fluctuations in the case of the field we use the
effective force

$\displaystyle\langle\delta X_{ee}^{2}\rangle=\frac{\partial\langle
X_{ee}\rangle}{\partial(\beta f_{\text{eff}})}=\frac{\partial\langle
X_{ee}\rangle}{\partial\epsilon}\left(\frac{\partial(\beta
f_{\text{eff}})}{\partial\epsilon}\right)^{-1}\;,$ (10)

where $f_{\text{eff}}$ is the effective force obtained from the MS equation
numerically. Equation 10 is solved numerically and plotted against the
dimensionless electric field in Figure 9 (B). In Figure 9 (A), for a uniform
tension, we find that the interpolated derivative function
$g(x)=\partial\langle X_{ee}\rangle/\partial x$, where $x=\beta fLl_{p}$ is
the force scaled to resemble the dimensionless field $\epsilon Ll_{p}$, agrees
with the simulations better than the MF model.

Figure 9: Comparison of the longitudinal fluctuations for WLCs in a field with
the literature –Normalized longitudinal fluctuations calculated using
interpolation method. (A) Interpolation equation $g(x)$ is evaluated as a
derivative of the end-to-end extension with respect to the homogeneous force
$x=\beta fLl_{p}$. The solid lines (black, blue, brown) correspond to 3
different values of the persistence length $l_{p}=2,15,30$. Solid dots with
error bars are simulations for $l_{p}=2$ (black), $l_{p}=15$ (blue), and
$l_{p}=30$ (brown). The predictions here agree better with the simulation than
the MF fluctuations in Figure 8. The finite size effects for low fields are
also shown. (B) The longitudinal fluctuations for an applied field can be
accurately computed by differentiating the theoretical extension (in Figure 5)
with respect to an effective force, $\beta
f_{eff}(\epsilon)=c(\epsilon)L\epsilon$. For low and high fields, the
effective forces reduce to $c(\epsilon)=1/2$ and 1/4 (respectively), shown in
the dashed red and blue respectively. The subdivided MF theory and the MS
theory with $c(\epsilon)=1/2$ agree well with the observed fluctuations.

The interpolated longitudinal fluctuations have a much better agreement than
the MF fluctuations for both tension and electric field because the extension
is accurate with the MF model and so extracting the fluctuation as a
derivative of the extension also gives a more accurate result. In Figure 9
(B), the longitudinal fluctuations are also compared to the MS plots for the
two values of the coefficients, namely, $f_{\text{eff}}\propto 1/4$ and
$f_{\text{eff}}\propto 1/2$. The former does badly for all ranges of the
electric field however the corrected effective force coefficient,
$c(\epsilon)=1/2$ is very close to the interpolated fluctuation curve. Since
the MF model predicts the FEC curves better than the fluctuations, so using
the slope of the FEC curves may be the reason that the interpolated functions
give more accurate statistics.

##  V Discussion and conclusion

In this paper, we have studied the statistics of charged worm-like chains
(WLCs) stretched under uniform electric fields. We have improved on an
existing mean-field theory Ha and Thirumalai (1995); Morrison and Thirumalai
(2009) used to predict the field-extension statistics for WLCs. We have shown
that a homogeneous mean-field theory fails to accurately capture the
statistics of the extension of the polymer. Such a homogeneous theory, with a
constant stretching parameter, leads to an overstretching of the bonds and the
model predicts a maximum extension of only $0.86L$ even when high electric
fields are applied. We show that the overstretching and the field-extension
plots can be corrected using the subdivided MF model which breaks the chain
into $n$ subdivisions with different stretching stiffness $\\{\lambda_{j}\\}$.
A variable stretching stiffness, imposed on the mean field level, ensures that
bonds that were overstretched in the homogeneous theory have a greater
resistance to stretching through the Lagrange multiplier $\lambda_{j}$, and
that bonds that were under stretched have a lower resistance to stretching.
From the field-extension curves, we also find that the effect of the
inhomogeneity in tension is prominent for high applied fields. In the
intermediate range, we find that the bonds closer to the tether ($s=0$) are
aligned more with the field compared to the free end ($s=L$). This is in
contrast to a uniform force stretching in optical tweezers experiments, where
at any point on the backbone of the chain the stretching force is the same.

The mean-field model is an analytically tractable approach, which means a wide
variety of experimental observables can also be calculated. Examples include
mean end-to-end extension, longitudinal fluctuations, or transverse
fluctuations. For transverse fluctuations, we recover the well-known stem-
flower or trumpet statistics, arising from an unequal stretching on the end of
the chain. The subdivided MF model accurately predicts the transverse
fluctuations in WLCs. Nanopore sequencing devices Heng et al. (2005); Qiu et
al. (2015, 2021); Lu et al. (2011) aim to increase the precision of base
sequence readings by decreasing the fluctuations while stretching DNA under
intermediate electric fields. Our calculations may help us understand the
molecular process of these systems better and may contribute towards improving
sequencing technology. In the case of longitudinal fluctuations, we find that
the isotropic mean-field approach fails to capture the longitudinal
fluctuations both in the case of tension or field. Since the FECs are
accurately predicted by the mean-field model, we propose an interpolation
method that generates the fluctuation plots using the slopes of the FECs. We
find that the fluctuations predicted by the interpolation method show a good
agreement with the simulation data.

Marko and Siggia’s Marko and Siggia (1995) well-known force-extension equation
has excellent agreement with the experimental force-extension curves for worm-
like chains stretched with uniform tension forces, $\beta f$, however, fails
to capture the correct statistics for fields. We find that simulations agree
better with the field-extension curves obtained from the subdivided MF model
in comparison to the MS equation for fields. The MS equation is derived in the
weakly bending rod limit, where the chain slightly fluctuates around the
stretching axis, or in other words, higher-order transverse fluctuations are
negligible. This assumption holds only for high electric fields when the chain
is highly stretched. For intermediate fields, we show that the transverse
fluctuations need a more exact evaluation due to the effect of the
inhomogeneous tension. Longitudinal fluctuations derived from the MS equation
with effective force, $f_{eff}=L\epsilon/2$ have good agreement with the
simulation and are close to the interpolated fluctuations obtained from the
derivative of the field-extension plots. The transverse fluctuations, however,
cannot be predicted using the MS model.

In this paper, we have considered a charged polymer with a uniform charge
density $\sigma$ but the model can be extended to more generalized cases of
variable charges such as in block co-polymers, where the sign of the charges
are alternate or there is a charge distribution, $\sigma(s)$, along the chain.
In this case, the effects of charge interactions on the backbone will be an
added term in the Hamiltonian and it is an interesting future direction for
this project. Our model does not account for screening charges explicitly but
it is an important effect for DNA in solutions, where the persistence length
is modified to be the electrostatic persistence length Skolnick and Fixman
(1977). The persistence length in our model can be implemented as an effective
length and we predict that the qualitative behavior of the statistics will not
change. While applications of this work in real systems may be in nanopore
sequencing devices but the effect of confinement Chen (2017) of the
nanochannels can lead to an interplay of stretching of the chain and
development of hair-pin-like bends Odijk (2006); Cifra et al. (2009) in the
chain. To explore this intriguing question significant progress in simulations
and theory is required but the ‘subdivided MF model’ can serve as a starting
point.

###### Acknowledgements.

This work was completed in part with resources provided by the Research
Computing Data Core at the University of Houston. We acknowledge funding from
the National Science Foundation, NSF-PHYS-2019745 supporting this work, as
well as computational resources through NSF-CNS-1338099.

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## Supplemental Information

### S1: Calculation of free energy for a pinned, charged WLC stretched in a
uniform electric field using MF model

To evaluate the free energy of a charged polymer ($\sigma>0$) in an electric
field, we take a path integral approach. The Hamiltonian in Equation 2 is
quadratic and can be evaluated using Gaussian integrals as follows.

$\mathcal{F}\propto-\log\left[\int\mathcal{D}[\mathbf{v}(s)]\exp(-\beta
H_{0})\right]\\\ =-\log\left[\int\mathcal{D}[\mathbf{v}(s)]\
\exp\left(-\frac{l_{0}}{2}\int_{0}^{L}ds\,\dot{\mathbf{v}}^{2}-\lambda\int_{0}^{L}ds\
\mathbf{v}^{2}\right)\exp(\mathbf{B}^{T}\cdot\mathbf{V}+\mathbf{C})\\\
\exp\left(-\delta_{0}\left(v_{x}(0)+\frac{\epsilon
L}{2\lambda}\right)^{2}-\delta_{0}\left(v_{y}(0)^{2}+v_{z}(0)^{2}\right)-\delta_{L}\mathbf{v}_{L}^{2}\right)\right]-\lambda
L/n-\delta_{0}-\delta_{L}\;.$

The quadratic terms in $\mathbf{v}(s)$ and $\dot{\mathbf{v}}(s)$ resemble the
Hamiltonian of a three-dimensional classical simple harmonic oscillator (SHO)
with mass, $m=l_{0}$ and frequency, $\Omega=\sqrt{\frac{2\lambda}{l_{0}}}$. We
can replace the quadratic part of the Hamiltonian with the simple harmonic
oscillator propagator between the endpoints $\mathbf{v}_{0}$ and
$\mathbf{v}_{L}$ and get

$\displaystyle K(\mathbf{v}_{0},\mathbf{v}_{L},\lambda,L)=\left(\frac{\Omega
l_{0}}{2\pi\sinh(\Omega L)}\right)^{3/2}\exp\left(-\Omega l_{0}\coth(\Omega
L)\frac{(\mathbf{v}_{0}^{2}+\mathbf{v}_{L}^{2})}{2}+\frac{\Omega
l_{0}}{\sinh(\Omega L)}\mathbf{v}_{0}\cdot\mathbf{v}_{L}\right).$ (S1)

Substituting the propagator in S1 results in the evaluation of a well-known
d-dimensional Gaussian integral

$\displaystyle\int\mathcal{D}[\mathbf{v}(s)]\ \exp(-\
\mathbf{V}^{T}\mathbf{A}\mathbf{V})\exp(\mathbf{B}^{T}\cdot\mathbf{V}+\mathbf{C})\propto\sqrt{\frac{\pi^{d}}{Det\mathbf{A}}}\exp\left(\frac{1}{4}\mathbf{B}^{T}\cdot\mathbf{A}^{-1}\cdot\mathbf{B}\right)\exp(\mathbf{C})\;.$

The transpose components of the $6\times 1$ dimensional coordinate matrix,
$\mathbf{V}^{T}=(v_{x}(0),v_{x}(L),\\\ v_{y}(0),v_{y}(L),v_{z}(0),v_{z}(L))$.
The coefficient matrix for the quadratic terms, $\mathbf{A}$, is tri-diagonal:

$\mathbf{A}=\left(\begin{array}[]{cccccc}a_{1,1}&a_{1,2}&&&&0\\\
a_{2,1}&a_{2,2}&a_{2,3}&&&\\\ &a_{3,2}&a_{3,3}&\ddots&&\\\
&&\ddots&\ddots&\ddots&a_{5,6}\\\ 0&&&&a_{6,5}&a_{6,6}\end{array}\right)\;,$

where odd diagonal terms are $a_{2i+1,2i+1}=\delta_{0}+(\Omega
l_{0}/2)\coth(\Omega L)$ and even diagonal terms are
$a_{2i,2i}=\delta_{L}+(\Omega l_{0}/2)\coth(\Omega L)$ for $i=0,1,2$. The off-
diagonal terms are $-\frac{\Omega l_{0}}{2\sinh(\Omega L)}$.

The transpose of the linear coefficient matrix $\mathbf{B}$’s is,

$\mathbf{B}^{T}=\left(-\frac{l_{0}\epsilon}{2\lambda}-\frac{\delta_{0}\epsilon
L}{\lambda}\,,\;\frac{l_{0}\epsilon}{2\lambda}\,,\;0\,,\;0\,,\;0\,,\;0\right)$

and finally the constant term matrix is
$\mathbf{C}=-\frac{\delta_{0}\epsilon^{2}L^{2}}{4\lambda^{2}}+\int_{0}^{L}ds\left(\frac{\epsilon^{2}(L-s)^{2}}{4\lambda}-\frac{l_{0}\epsilon^{2}}{8\lambda^{2}}\right)$.

Figure 10: FIG S1: Power laws of stretching parameters as a function of the
electric field –Scalings of the Lagrange multipliers $\delta_{0,L}$ and
$\lambda$ as a function of the electric field $\epsilon=\sigma\beta E$. (A)
The low field values for $\delta_{0,L}$ are close to the zero force or field
values derived in Ha et al. Ha and Thirumalai (1995), $\delta=0.75$. The
asymptotic scaling is proportional to $\epsilon^{1/2}$. The difference in the
sign of the $\delta$ parameters indicate the difference in the endpoint
conditions. (B) Low field value for the $\lambda$ stretching parameter is
close to the zero field value of $\lambda_{0}=9/8l_{0}=0.05$ for
$l_{0}=3l_{p}/2$ and $l_{p}=15$. The intermediate field region scaling for
$\lambda$ is a mix of power laws between $\epsilon^{0}$ and $\epsilon^{1}$,
which is the asymptotic limit.

Plugging all the matrices into the d-dimensional Gaussian integral gives the
partition function and the free energy. However, calculating the determinant
of the tri-diagonal matrix is formidable and for long semiflexible chains, we
take the assumption to the first order $\coth(\Omega L)\sim 1$ and
$1/\sinh(\Omega L)\sim 0$. The assumption fails for stiff polymer chains with
persistence length $l_{0}>>L$. This assumption reduces $\mathbf{A}$ to a
diagonal matrix whose determinant is easily calculated as the product of all
the elements. The diagonal terms of the matrix are $\delta_{0}+\Omega
l_{0}/2$, or $\delta_{L}+\Omega l_{0}/2$, depending on the odd or even
position. The coefficient of the propagator, $\left(\frac{\Omega
l_{0}}{2\pi\sinh(\Omega L)}\right)^{3/2}$ expanded to the second order for the
same assumption results in $\left(\frac{\Omega l_{0}e^{-\Omega
L}}{\pi}\right)^{3/2}$.

To obtain the scaling of the Lagrange multipliers, the mean-field equations
are solved numerically. From the plots in Figure S1, we deduce from the fits
that $\lambda$ scales linearly with $\epsilon$, and the end-point Lagrange
multipliers $\delta_{0}$ and $\delta_{L}$ scale as $\sqrt{\epsilon}$. Plugging
these scalings back into the free energy gives us the coefficients of the
scaling laws, $\lambda\sim L\epsilon/2\sqrt{3}$, $\delta_{0}\sim
0.341\sqrt{Ll_{p}\epsilon}$, and $\delta_{L}\sim-0.465\sqrt{Ll_{p}\epsilon}$,
where $l_{p}$ is the true persistence length of the chain.

## S2: Calculation of free energy for a pinned, charged WLC stretched in a
uniform electric field using the subdivided MF model

The free energy of the subdivided MF Hamiltonian in Equation 4 is calculated
using Gaussian integrals. We complete squares in Equation 3 using the
transformation for the $j^{th}$ subdivision,
$\mathbf{v}_{j}=(v_{xj},v_{yj},v_{zj})=(u_{xj}-(L-s)\epsilon/2\lambda_{j},u_{yj},u_{zj})$.
The free energy is

$\mathcal{F}_{n}\propto-\log\left[\int\mathcal{D}[\mathbf{v}(s)]\exp(-\beta
H_{0}[\mathbf{v}_{j}(s)])\right]\\\
=-\log\left[\int\mathcal{D}[\mathbf{v}(s)]\
\left\\{\sum_{j=1}^{n}\exp\left(-\frac{l_{0}}{2}\int_{j}ds_{j}\,\dot{\mathbf{v}_{j}}^{2}-\lambda_{j}\int_{j}ds_{j}\
\mathbf{v}_{j}^{2}\right)\right\\}\exp(\mathbf{B}_{n}^{T}\cdot\mathbf{V}_{n}+\mathbf{C}_{n})\\\
\exp\left(-\delta_{0}\left(v_{x}(0)+\frac{\epsilon
L}{2\lambda_{1}}\right)^{2}-\delta_{0}\left(v_{y}(0)^{2}+v_{z}(0)^{2}\right)-\delta_{L}\mathbf{v}_{L}^{2}\right)\right]-\sum_{j=1}^{n}\lambda_{j}L/n-\delta_{0}-\delta_{L}$

, where the sum over all the $j=1,...,n$ subdivisions give the total path
integral of the chain, $\mathbf{B}_{n}$ is the coefficient matrix for the
linear terms, and $\mathbf{C}_{n}$ is the constant term matrix, $\lambda_{j}$
is the stretching parameter for the $j^{th}$ subdivision, and $\lambda_{1}$ is
the stretching parameter of the first subdivision attached to the pinned end
of the chain. Just as in the undivided chain case, the simple harmonic
oscillator propagator can replace the quadratic terms in the curly braces, and
for each subdivision, the propagator with frequency,
$\Omega_{j}=\sqrt{2\lambda_{j}/l_{0}}$, is given by

$K(\mathbf{v}_{j-1},\mathbf{v}_{j},\lambda_{j},L/n)=\left(\frac{\Omega_{j}l_{0}}{2\pi\sinh(\Omega_{j}L/n)}\right)^{3/2}\exp\left(-\Omega_{j}l_{0}\coth(\Omega_{j}L/n)\frac{(\mathbf{v}_{j-1}^{2}+\mathbf{v}_{j}^{2})}{2}\\\
+\frac{\Omega_{j}l_{0}}{\sinh(\Omega_{j}L/n)}\mathbf{v}_{j-1}\cdot\mathbf{v}_{j}\right).$

For a chain with $n$ subdivisions, the coefficients of the quadratic terms
form the $3(n+1)\times 3(n+1)$ dimensional $\mathbf{A}_{n}$ matrix as follows,

$\mathbf{A}_{n}=\left(\begin{array}[]{cccccc}a_{1,1}&a_{1,2}&&&&0\\\
a_{2,1}&a_{2,2}&a_{2,3}&&&\\\ &a_{3,2}&a_{3,3}&\ddots&&\\\
&&\ddots&\ddots&\ddots&a_{3(n+1)-1,3(n+1)}\\\
0&&&&a_{3(n+1),3(n+1)-1}&a_{3(n+1),3(n+1)}\end{array}\right)\;,$

and the points of subdivision spaced at every $L/n$ arc length are denoted by
the $3(n+1)\times 1$ dimensional matrix, $\mathbf{V}_{n}$ matrix whose
transpose components are,
$\mathbf{V}^{T}_{n}$$\equiv(v_{x0},v_{x1},...,v_{xj},...,\\\
v_{xn},v_{y0},v_{y1},...,v_{yj},...,v_{yn},v_{z0},v_{z1},...,v_{zj},...,v_{zn})$.
We take the assumption of long semiflexible chains with a large number of
subdivisions and retain only diagonal terms in the $\mathbf{A}_{n}$ matrix.
The diagonal terms corresponding to the endpoints $\mathbf{v}(0)$ and
$\mathbf{v}(L)$ are, $\delta_{0}+\Omega_{1}l_{0}/2$ and
$\delta_{L}+\Omega_{n}l_{0}/2$ respectively. For the internal points denoting
the points of subdivision, the diagonal terms are
$l_{0}(\Omega_{j-1}+\Omega_{j})/2$. For a diagonal matrix, calculating the
determinant is straightforward and it is the product of all the diagonal
elements.

The linear terms form the $3(n+1)\times 1$ dimensional $\mathbf{B}_{n}$ matrix
and the transpose components of $\mathbf{B}_{n}$ terms are given by,

$\mathbf{B}_{n}^{T}=\left(\begin{array}[]{c}-\frac{l_{0}\epsilon}{2\lambda_{1}}-\frac{\delta_{0}\epsilon
L}{\lambda_{1}},\frac{l_{0}\epsilon}{2\lambda_{1}}-\frac{l_{0}\epsilon}{2\lambda_{2}},\ldots,\frac{l_{0}\epsilon}{2\lambda_{j-1}}-\frac{l_{0}\epsilon}{2\lambda_{j}},\ldots,\frac{l_{0}\epsilon}{2\lambda_{n}},0,0,\ldots,0\end{array}\right)\;,$

where only the terms in the direction of the field (here x-direction) are non-
zero and all other terms are zero.

The one dimensional constant matrix,
$\mathbf{C}_{n}=-\frac{\delta_{0}\epsilon^{2}L^{2}}{4\lambda_{1}^{2}}+\sum_{j=1}^{n}\int_{j}ds_{j}\left(\frac{\epsilon^{2}(L-s_{j})^{2}}{4\lambda_{j}}-\frac{l_{0}\epsilon^{2}}{8\lambda_{j}^{2}}\right)$,
where $\int_{j}\equiv=\int_{(j-1)L/n}^{jL/n}$.

The partition function of the subdivided MF Hamiltonian can now be evaluated
using the Gaussian integral formula in Equation S1 and by replacing
$\mathbf{A},\mathbf{B},\mathbf{C}$ with the new matrices
$\mathbf{A}_{n},\mathbf{B}_{n},\mathbf{C}_{n}$. The free energy follows from
the partition function,
$\mathcal{F}_{n}=-\log\left[\int\mathcal{D}[\mathbf{v}(s)]\exp(-\beta
H_{0}[\mathbf{v}_{j}(s)])\right]-\sum_{j=1}^{n}\lambda_{j}L/n-\delta_{0}-\delta_{L}$.
The mean-field equations
$\partial\mathcal{F}_{n}/\partial\lambda_{j}=\partial\mathcal{F}_{n}/\partial\delta_{0}=\partial\mathcal{F}_{n}/\partial\delta_{L}=0$
give the distribution of the stretching parameters.

The recipe outlined in this section calculates the distribution of stretching
parameters, $\lambda_{j}$. For a zero electric field, the stretching
parameters are all equal and are that of a WLC without any stretching forces,
$\lambda=9/8l_{0},\delta=3/4$. For non-zero values of the electric field, the
mean-field equations can be solved numerically. For long WLCs in the high
electric field region, the scaling of the stretching parameters can be
evaluated analytically. From the undivided chain scaling, $\lambda\sim
L\epsilon/2\sqrt{3}$ we obtain that the stretching parameters depend on the
electric field linearly, and so it is safe to assume that $\lambda_{j}\sim
c_{j}\epsilon$ with coefficients, $c_{j}$, depending on the subdivision’s
location on the chain. In the high electric field limit and plugging in this
scaling of $\lambda_{j}$’s in the free energy $\mathcal{F}_{n}$ we conclude
that the leading order terms in the free energy are,
$\mathcal{F}_{n}\sim\sum_{j=1}^{n}\int_{j}ds_{j}\frac{\epsilon^{2}(L-s_{j})^{2}}{4\lambda_{j}}-\lambda_{j}L/n$.
The mean-field equation, $\partial\mathcal{F}/\partial\lambda_{j}=0$ results
in,
$\lambda_{j}=\left(\int_{j}ds_{j}(L-s_{j})^{2}\epsilon^{2}n/4L\right)^{1/2}$,
for the $j^{th}$ subdivision’s stretching parameter and so on. For a large
number of subdivisions and vanishing $L/n$ length of the subdivisions, the
$\lambda_{j}$ scaling for high electric field simplifies to
$\lambda(s)\sim(L-s)\epsilon/2$ as shown in Figure 4.

## S3: Calculation of a mean end-to-end extension using MF approach

To calculate the ensemble average of any quantity, $\langle g\rangle$ using
the definition of mean from statistical mechanics, $\langle
g\rangle=\int\mathcal{D}[\mathbf{v}(s)]\ g\exp(-\beta
H_{0})/\int\mathcal{D}[\mathbf{v}(s)]\exp(-\beta H_{0})$. The average end-to-
end extension in the presence of an electric field is the total of all the
bond vectors and the ensemble average for this quantity is, $\langle
g\rangle=\langle\int_{0}^{L}dsu_{x}(s)\rangle=\left\langle\sum_{j=1}^{n}\int_{j}ds_{j}u_{x}(s_{j})\right\rangle$.
To make the calculation of this ensemble average simpler, we add uniform
auxiliary forces Ha and Thirumalai (1995), $\alpha_{j}$’s to the Hamiltonian
in Equation 3 –

$\displaystyle\beta H_{0}[\mathbf{u}_{j}(s)]$
$\displaystyle=\delta_{0}\mathbf{u}_{0}^{2}+\delta_{L}\mathbf{u}_{L}^{2}+\frac{l_{0}}{2}\int_{0}^{L}ds\dot{\mathbf{u}}^{2}+\sum_{j=1}^{n}\lambda_{j}\int_{j}ds_{j}\mathbf{u}_{j}^{2}$
$\displaystyle-\boldsymbol{\epsilon}\cdot\int_{0}^{L}ds\int_{0}^{s}ds^{\prime}\mathbf{u}(s^{\prime})-\sum_{j=1}^{n}\alpha_{j}\int_{j}ds_{j}u_{x}(s_{j}),$
(S2)

and complete squares as before. All the auxiliary fields are positive Lagrange
multipliers and the index $j$ has been added to identify the extension
relative to the $j^{th}$ subdivision. For an electric field applied in the
x-direction, the new transformation in coordinates to complete the squares
including the auxiliary fields is,
$w_{xj}=(u_{xj}-((L-s)\epsilon+\alpha_{j})/2\lambda_{j},u_{yj},u_{zj})$. The
Hamiltonian in Equation S2 becomes quadratic with this transformation and the
partition function is,
$\exp(-\mathcal{F}_{n}^{\alpha})=\exp(\log\left[\mathcal{D}[\mathbf{w}(s)]\exp(-\beta
H_{0}[\mathbf{w}_{j}(s)])\right]-\sum_{j=1}^{n}\lambda_{j}L/n-\delta_{0}-\delta_{L})$,
where $\mathcal{F}_{n}^{\alpha}$ is the free energy that is calculated using
Gaussian integrals. The coefficient matrices can be derived as follows.

The quadratic coefficient matrix, $\mathbf{A}_{n}=\mathbf{A}_{n}^{\alpha}$ is
unaffected by the transformation and retains all the terms for the coordinate
matrix $\mathbf{W}_{n}^{T}$. The assumption of a long chain retains the
diagonal terms in $\mathbf{A}_{n}$ as before and the product of all the
elements is the determinant of the matrix. The other two matrices correspond
to the linear and constant terms change and have the auxiliary fields
included. For the linear terms only the endpoint terms the effect of the
auxiliary fields and all other terms remain the same as $\mathbf{B}_{n}^{T}$.
The endpoint terms for $w_{x}(s=0)$ and $w_{x}(s=L)$ are respectively,
$-\frac{l_{0}\epsilon}{2\lambda_{1}}-\frac{\delta_{0}(\alpha_{1}+\epsilon
L)}{\lambda_{1}}$ and
$-\frac{\delta_{L}\alpha_{n}}{\lambda_{n}}+\frac{l_{0}\epsilon}{2\lambda_{n}}$.
The constant term matrix changes as,
$\mathbf{C}_{n}^{\alpha}=-\frac{\delta_{0}(\epsilon
L+\alpha_{1})^{2}}{4\lambda_{1}^{2}}-\frac{\delta_{L}\alpha_{n}^{2}}{4\lambda_{n}^{2}}+\sum_{j=1}^{n}\int_{j}ds_{j}\left(\frac{(\epsilon(L-s_{j})+\alpha_{j})^{2}}{4\lambda_{j}}-\frac{l_{0}\epsilon^{2}}{8\lambda_{j}^{2}}\right)$.

The advantage of adding auxiliary fields, $\alpha_{j}$ to the Hamiltonian, is
we can now calculate averages easily, like the end-to-end distance, which is
given by
$\sum_{j=1}^{n}-\frac{\partial\mathcal{F}_{n}^{\alpha}}{\partial\alpha_{j}}\Big{|}_{\alpha_{j}=0}$.
The extension for $n$ subdivisions is calculated in the main text. We should
note that the distribution in the stretching parameters, $\lambda_{j}$, does
not change because the distribution is calculated from the same mean-field
equations as our subdivided MF model,
$\frac{\partial\mathcal{F}_{n}^{\alpha}}{\partial\lambda_{j}}=0\Big{|}_{\alpha_{j}=0}$.

## S4: Calculation of transverse fluctuations using the subdivided MF model

To evaluate the transverse fluctuations from the subdivided chain Hamiltonian
in Equation 8, we recognize that the calculations do not need to account for
the electric field term explicitly as the perpendicular direction only feels
the effect of the field through the stretching parameters. The stretching
parameters have already been evaluated for the subdivided chain and for any
value of the electric field the stretching parameters can be extracted and
plugged into the transverse fluctuations. The transformed Hamiltonian from
Equation 8 with the following transformation for arc length, $s=kL/n$ is given
by $\mathbf{w}^{\perp}_{j}=\mathbf{u}^{\perp}_{j}-\alpha/2\lambda_{j}$ is

$\displaystyle\beta H_{0}[\mathbf{w}^{\perp}_{j}(s)]$
$\displaystyle=\delta_{0}(\mathbf{w}_{0}^{\perp}+\alpha/2\lambda_{1})^{2}+\delta_{L}(\mathbf{u}_{L}^{\perp})^{2}+\frac{l_{0}}{2}\int_{0}^{s}ds^{{}^{\prime}}(\dot{\mathbf{w}}^{\perp})^{2}+\sum_{j=1}^{k}\lambda_{j}\int_{j}ds^{{}^{\prime}}_{j}(\mathbf{w}_{j}^{\perp})^{2}$
$\displaystyle-\sum_{j=1}^{k}\int_{0}^{s}ds^{{}^{\prime}}\frac{\alpha^{2}}{4\lambda_{j}}+\frac{l_{0}}{2}\int_{s}^{L}ds^{{}^{\prime}}(\dot{\mathbf{u}}^{\perp})^{2}+\sum_{j={k+1}}^{n}\lambda_{j}\int_{j}ds^{{}^{\prime}}_{j}(\mathbf{u}_{j}^{\perp})^{2}.$
(S3)

The harmonic oscillator propagator from this transformed Hamiltonian is used
to calculate the transverse fluctuations at every point of subdivision or
$s=kL/n$, where $k=1,2,...,n$ is proportional to,
$\left(\prod_{j=1}^{k}K(\mathbf{w}^{\perp}_{j-1},\mathbf{w}^{\perp}_{j},\lambda_{j},L/n)\right)\left(\prod_{j=k+1}^{j=n}K(\mathbf{u}^{\perp}_{j-1},\mathbf{u}^{\perp}_{j},\lambda_{j},L/n)\right)$.
Since the transverse fluctuations are calculated at internal points, $s$, the
argument that the differences at the points of subdivision,
$\eta_{j}=1/\lambda_{j+1}-1/\lambda_{j}$ become small does not give the
correct statistics. The propagator expression, therefore, has to be replaced
by the same coordinates for continuity, i.e. $\mathbf{u}$ and the propagator
is then equivalent to
$\left(\prod_{j=1}^{k}K(\mathbf{u}^{\perp}_{j-1}-\alpha/2\lambda_{j},\mathbf{u}^{\perp}_{j}-\alpha/2\lambda_{j},\lambda_{j},L/n)\right)\left(\prod_{j=k+1}^{j=n}K(\mathbf{u}^{\perp}_{j-1},\mathbf{u}^{\perp}_{j},\lambda_{j},L/n)\right)$.
Using this propagator from the harmonic terms and along with the additional
terms in the Hamiltonian Equation 8, the free energy is evaluated using
Gaussian integrals as before. The transverse fluctuations is finally given by
$\partial^{2}(-\mathcal{F}^{\perp})/\partial\alpha^{2}\Big{|}_{\alpha=0}$.

## S5: Calculation of effective force from the MS model for WLCs stretched in
electric fields

Figure 11: FIG S2: Electric field as an effective tension force –The effective
force derived from the MS equation is plotted against the field. The low and
the high field values are constant but the intermediate region has a non-
linear variation with the field.

Marko et al. Marko and Siggia (1995) proposed that the electric field can be
replaced using an effective tension $f_{\text{eff}}$ from the MS equation,

$\displaystyle\beta fl_{p}=\frac{\langle
X_{ee}\rangle}{L}+\frac{1}{4(1-\langle X_{ee}\rangle)^{2}}-\frac{1}{4}.$ (S4)

Plugging in the analytical expression for the total extension $\langle
X_{ee}\rangle$ (obtained using the subdivided chain model equations 6) into
the MS Equation S4 we can solve for $\beta f_{\text{eff}}$ numerically as
shown in Figure S2.