# Mechanical response to tension and torque of molecular chains via
statistically interacting particles associated with extension, contraction,
twist, and supercoiling

Aaron C. Meyer  Department of Physics, University of Rhode Island, Kingston RI
02881, USA  Michael Karbach  Fachgruppe Physik, Bergische Universität
Wuppertal, D-42097 Wuppertal, Germany  Ping Lu  Department of Physics, Stetson
University, DeLand, FL 32723, USA  Gerhard Müller  Department of Physics,
University of Rhode Island, Kingston RI 02881, USA

###### Abstract

A methodology for the statistical mechanical analysis of polymeric chains
under tension introduced previously is extended to include torque. The
response of individual bonds between monomers or of entire groups of monomers
to a combination of tension and torque involves, in the framework of this
method of analysis, the (thermal or mechanical) activation of a specific mix
of statistically interacting particles carrying quanta of extension or
contraction and quanta of twist or supercoiling. The methodology, which is
elucidated in applications of increasing complexity, is capable of describing
the conversion between twist chirality and plectonemic chirality in
quasistatic processes. The control variables are force or extension and torque
or linkage (a combination of twist and writhe). The versatility of this
approach is demonstrated in two applications relevant and promising for
double-stranded DNA under controlled tension and torque. One application
describes conformational transformations between (native) B-DNA, (underwound)
S-DNA, and (overwound) P-DNA in accord with experimental data. The other
application describes how the conversion between a twisted chain and a
supercoiled chain accommodates variations of linkage and excess length in a
buckling transition.

##  I Introduction

This work investigates the statistical mechanics of molecular chains with
(internal and external) torsional constraints. The bonds between monomers are
not rotatable, neither freely nor across periodic energy barriers. In
consequence, the molecular chain responds to torque by a buildup of torsional
elastic energy, which is necessarily coupled to elastic energy associated with
elongation or contraction. Hence torque and tension both vary, in general,
when the molecular chain is being twisted or stretched. Research on double-
stranded (ds) DNA is the most notable field of applications by far MS94 ; MS95
; SAB+96 ; Mark97 ; Mark98 ; ABLC98 ; Smit98 ; SABC98 ; HYZ99 ; SAB+99 ;
LRS+99 ; BM00 ; ZZY00 ; SABC00 ; SACB00 ; SLCM01 ; BBS03 ; BSG+03 ; Neuk04 ;
LJL+06 ; Mark07 ; FDS+08 ; DFS+09 ; SW09 ; WRC09 ; NR11 ; GLO+11 ; OK11 ; OK12
; MN13 ; ELS13 .

Experimentally, the ends of the chain are mounted to the measuring device such
that the total angle of twist plus writhe (named linkage) is either held
constant or is controllably changed. In some instances, the application of
torque (or tension) nucleates conformational transformations. One conformation
then grows at the expense of another. In other instances, a coupled variation
of torque and tension, initiates and grows supercoiling, converting twist into
writhe and changing the excess length simultaneously in the process.

In this setup, it is necessary to consider a mechanical agent which
simultaneously exerts tension $J$ and torque $\tau$ in some combination. The
molecular chain responds with a combination of extension/contraction length
$L$ and twist/writhe angle $\phi$. The statistical mechanical analysis,
therefore, deals with three pairs of conjugate thermodynamic variables,
$(J,L)$, $(\tau,\phi)$, and $(T,S)$, where the last pair are temperature and
entropy.

Our method of analysis leads, by default, to a Gibbs free energy with natural
independent variables $T,J,\tau$, from which the thermal and mechanical
responses are derived via derivatives. Thermodynamic relations are readily
inverted to produce relations amenable to direct comparisons with experimental
data. This method of analysis was previously introduced in some detail for
applications to molecular chains under tension without torsional constraints
mct1 .

Recent advances in single-molecule biophysics, specifically the enormous
progress achieved in the experimental investigation of DNA double helices
subject to controllable tension and torque, are in need of ever more versatile
theoretical approaches for the interpretation of new data. This paper offers a
contribution with a demonstration of its merits to satisfy that need. It
comprises the natural extension of a general methodology known as fractional
exclusion statistics (FES) Hald91a ; Wu94 ; Isak94 ; Anghel ; NA14 ; LVP+08 ;
copic ; picnnn ; pichs .

The approach has already proven its usefulness in numerous quantum and
classical applications LMK09 ; PMK07 ; sivp ; GKLM13 ; janac2 ; cohetra ,
notably for ds-DNA subjected to torsionally unconstrained stretching mct1 .
The results of this approach aligned with experimental data of the force-
extension characteristic across regimes of entropic elasticity (thermal
umbending), enthalpic elasticity (beyond contour length), and an
overstretching transition.

The technical aspects of the statistical mechanical analysis, which is firmly
grounded in quasistatic processes, are briefly reviewed in Sec. II with
emphasis on those aspects that are in need of eloborations beyond the account
given in Ref. mct1 . One elementary application, which illustrates the
coupling between the two pairs $(J,L)$ and $(\tau,\phi)$ of conjugate
thermodynamic variables in the context of torsionally constrained twisting and
stretching, is worked out in Sec. III.

The focus of Sec. IV is on transitions between three well established
conformations of ds-DNA. We infer from a single partition function the
empirically found phase diagram with three well-defined boundaries between
(native) B-DNA, (underwound) S-DNA, and (overwound) P-DNA. The coexistence
between twisted chain and supercoiled chain described within the same
framework is the topic of Sec. V. The distinction is made between high-tension
supercoils (rope variety) and low-tension supercoils (garden-hose variety),
analyzed in Secs. VI and VII, respectively. The former is set up for the
demonstration in Sec. VIII of a buckling transition.

##  II Methodology

Quasistatic processes operate within the realm of equilibrium statistical
mechanics. The response of a thermodynamic system (here a molecular chain
embedded in a fluid) to agents of change (here sources of tension and torque)
is described by thermodynamic relations derived from a partition function. The
partition function is a sum over microstates of terms weighted according to
energies. Microstates are characterized by their quasiparticle content.
Quasiparticles are (thermally or mechanically) activated from some reference
state.

For a system of statistically interacting particles from species
$m=1,\ldots,M$, the partition function,

$Z=\sum_{\\{N_{m}\\}}W(\\{N_{m}\\})e^{-\beta
E(\\{N_{m}\\})},\quad\beta\doteq\frac{1}{k_{\mathrm{B}}T},$ (1)

depends on the multiplicity of microstates,

$\displaystyle W(\\{N_{m}\\})$
$\displaystyle=\prod_{m=1}^{M}\left(\begin{array}[]{c}d_{m}+N_{m}-1\\\
N_{m}\end{array}\right),$ (2c) $\displaystyle d_{m}$
$\displaystyle=A_{m}-\sum_{m^{\prime}=1}^{M}g_{mm^{\prime}}(N_{m^{\prime}}-\delta_{mm^{\prime}}),$
(2d)

and their energies,

$E(\\{N_{m}\\})=E_{\mathrm{pv}}+\sum_{m=1}^{M}N_{m}\epsilon_{m},$ (3)

where $E_{\mathrm{pv}}$ is the energy of the reference state (pseudo-vacuum)
and the $\epsilon_{m}$ are the particle activation energies. Multiplicity $W$
and energy $E$ are functions of particle content $\\{N_{m}\\}$. The
statistical interactions between particles are encoded in arrays of (non-
negative, rational) capacity constants $A_{m}$ and (rational) statistical
interaction coefficients $g_{mm^{\prime}}$.

The keystone in this scheme is the generalized Pauli principle, of which (2d)
is an integrated version. It was proposed and first used by Haldane Hald91a .
The evaluation of (1) was investigated by Wu Wu94 , Isakov Isak94 , Anghel
Anghel , and others LMK09 ; PMK07 ; sivp  at various levels of generality. For
a macroscopic system, we can write the partition function in the form,

$Z=\prod_{m=1}^{M}\big{(}1+w_{m}^{-1}\big{)}^{A_{m}},$ (4)

where the (real, positive) $w_{m}$ are solutions of the coupled nonlinear
algebraic equations,

$e^{\beta\epsilon_{m}}=(1+w_{m})\prod_{m^{\prime}=1}^{M}\big{(}1+w_{m^{\prime}}^{-1}\big{)}^{-g_{m^{\prime}m}}.$
(5)

The average numbers $\langle N_{m}\rangle$ of particles from all species are
the solutions, for given $\\{w_{m}\\}$, of the coupled linear equations,

$w_{m}\langle N_{m}\rangle+\sum_{m^{\prime}=1}^{M}g_{mm^{\prime}}\langle
N_{m^{\prime}}\rangle=A_{m}.$ (6)

In this work, all particles carry quanta of length and angle. Their activation
energies are of the general form,

$\epsilon_{m}=\gamma_{m}-JL_{m}-\tau\phi_{m}.$ (7)

The quantum of length $L_{m}$ is positive for extension particles and negative
for contraction particles. Likewise, the quantum of angle $\phi_{m}$ can be
positive or negative. In DNA applications, particles with positive $\phi_{m}$
overwind the double helix and particles with negative $\phi_{m}$ underwind it
when activated. The angle $\phi_{m}$ represents twist in some particle species
and writhe in others. Twist particles and writhe particles typically have
quite different length quanta $L_{m}$. The role of the energy constant
$\gamma_{m}$ varies between species and applications.

We infer from the partition function (4) the Gibbs free energy,

$G(T,J,\tau,N)=-k_{\mathrm{B}}T\ln Z,$ (8)

where the dependence on $T$ comes from $\beta$ in (5), the dependence on $J$,
$\tau$ from (7) via (5) and the dependence on $N$ (the number of bonds) is
hidden in the capacity constant $A_{m}$ of one or several particle species in
a way that guarantees thermodynamic extensivity of $G$. The quantities of
primary interest here are entropy, excess length, and linkage, obtained via
partial derivatives,

$S=-\frac{\partial G}{\partial T},\quad\langle L\rangle=-\frac{\partial
G}{\partial J},\quad\langle\phi\rangle=-\frac{\partial G}{\partial\tau}.$ (9)

These thermodynamic functions can also be inferred from particle population
averages via Eqs. (6) mct1 ; Isak94 ; sivp :

$\displaystyle S=k_{\mathrm{B}}\sum_{m=1}^{M}\Big{[}\big{(}\langle
N_{m}\rangle+{Y}_{m}\big{)}\ln\big{(}\langle N_{m}\rangle+{Y}_{m}\big{)}$
$\displaystyle\hskip 56.9055pt-\langle N_{m}\rangle\ln\langle
N_{m}\rangle-{Y}_{m}\ln{Y}_{m}\Big{]},$ (10a)
$\displaystyle{Y}_{m}\doteq{A}_{m}-\sum_{m^{\prime}=1}^{M}g_{mm^{\prime}}\langle
N_{m^{\prime}}\rangle,$ (10b) $\langle L\rangle=\sum_{m=1}^{M}L_{m}\langle
N_{m}\rangle,\quad\langle\phi\rangle=\sum_{m=1}^{M}\phi_{m}\langle
N_{m}\rangle.$ (11)

In Ref. mct1  we explained the different categories of particles: compact
particles, which exist side by side, and nested particles, which form
hierarchical structures. Particles at level 1 modify individual bonds and
particles at level 2 modify entire segments of the molecular chain. We also
worked out general solutions for several sets of particles. None of this will
be reiterated here. Instead we will comment on all essentials in the context
of each application with pointers to prior work.

##  III Twist contraction

The first application pertains to an idealized double-stranded molecular chain
whose reference state is a ladder with no native helical structure. The
applied tension $J$ is assumed to remain below the threshold of significant
contour elasticity such as discussed in Sec. III of mct1 . If a torque $\tau$
is applied the ladder conformation responds by a combination of twist and
contraction.

We model this response with a system of two species of particles that carry
quanta of twist angle and contraction length. The combinatorial specifications
of these particles are compiled in Table 1. They are level-1 compacts as
introduced in Sec. II.E of Ref. mct1  along with a general solution. The
particle activation energies have the general form (7), which we rewrite as

$\epsilon_{\pm}=\gamma_{\mathrm{t}}+JL_{\mathrm{t}}\mp\tau\phi_{\mathrm{t}}.$
(12)

The three energetic specifications are an elastic energy constant
$\gamma_{\mathrm{t}}>0$, a quantum of contraction length $L_{\mathrm{t}}>0$,
and a quantum of twist angle $\phi_{\mathrm{t}}>0$. The two particle species
account for the two senses of torque and twist, both of which are associated
with a contraction.

Table 1: Capacity constants $A_{m}$ and statistical interaction coefficients
$g_{mm^{\prime}}$ for the two species of level-1 compacts.

$m$ |  $A_{m}$
---|---
$+$ |  $N-1$
$-$ |  $N-1$

$g_{mm^{\prime}}$ |  $+$ |  $-$
---|---|---
$+$ | $1$ | $1$
$-$ | $0$ | $1$

If a positive torque $\tau$ is applied at constant tension $J$, the activation
energy $\epsilon_{+}$ decreases, whereas $\epsilon_{-}$ increases. The effect
is a positive twist and a contraction. Likewise, a negative torque favors the
activation of particles with activation energies $\epsilon_{-}$, which
produces a negative twist and a contraction again. On the other hand, if we
increase $J$ at constant (positive or negative) $\tau$, then both activation
energies $\epsilon_{\pm}$ increase. Twist particles from both species are
gradually frozen out. As the chain untwists, its contraction diminishes.

The Gibbs free energy per bond of a long chain, $\bar{G}(T,J,\tau)$, inferred
from the solution of Eqs. (5),

$w_{+}=e^{\beta\epsilon_{+}},\quad
w_{-}=e^{\beta\epsilon_{-}}\big{(}1+e^{-\beta\epsilon_{+}}\big{)}$ (13)

via (4) and (8), becomes

$\bar{G}\doteq\lim_{N\to\infty}\frac{G}{N}=-k_{\mathrm{B}}T\ln\Big{(}1+e^{-\beta\epsilon_{+}}+e^{-\beta\epsilon_{-}}\Big{)}.$
(14)

The first partial derivatives (9),

$\displaystyle\bar{S}\doteq\lim_{N\to\infty}\frac{S}{Nk_{\mathrm{B}}}$
$\displaystyle=\Bigg{[}\ln\Big{(}1+e^{-\beta\epsilon_{+}}+e^{-\beta\epsilon_{-}}\Big{)}$
$\displaystyle\hskip
14.22636pt+\frac{\beta\epsilon_{+}e^{-\beta\epsilon_{+}}+\beta\epsilon_{-}e^{-\beta\epsilon_{-}}}{1+e^{-\beta\epsilon_{+}}+e^{-\beta\epsilon_{-}}}\Bigg{]},$
(15a) $\bar{L}\doteq\lim_{N\to\infty}\frac{\langle
L\rangle}{N}=-L_{\mathrm{t}}\,\frac{e^{-\beta\epsilon_{+}}+e^{-\beta\epsilon_{-}}}{1+e^{-\beta\epsilon_{+}}+e^{-\beta\epsilon_{-}}},$
(15b)
$\bar{\phi}\doteq\lim_{N\to\infty}\frac{\langle\phi\rangle}{N}=\phi_{\mathrm{t}}\,\frac{e^{-\beta\epsilon_{+}}-e^{-\beta\epsilon_{-}}}{1+e^{-\beta\epsilon_{+}}+e^{-\beta\epsilon_{-}}},$
(15c)

represent entropy, (negative) extension length, and twist angle, respectively.

Figure 1: (a) Contraction distance $\hat{L}$, (b) twist angle $\hat{\phi}$,
and (c) entropy $\hat{S}$, all versus tension $\hat{J}$ and torque
$\hat{\tau}$ at constant temperature $\hat{T}=1$. The nine contour lines from
dark to bright in each panel are at (a) $\hat{L}=0.098,\ldots,0.882$, (b)
$\hat{\phi}=-0.76,\ldots,0.76$, and (c) $\hat{S}=0.097,\ldots,0.873$. Panels
(d)-(f) show the same quantities plotted versus $\hat{\tau}$ at constant
$\hat{J}=2.5$ and $\hat{T}=0.5,1,2$.

The contour plots in Fig. 1 visualize (in scaled units) the dependence of
contraction distance $\hat{L}\doteq|\bar{L}|/L_{\mathrm{t}}$, twist angle
$\hat{\phi}\doteq\bar{\phi}/\phi_{\mathrm{t}}$, and entropy
$\hat{S}\doteq\bar{S}/k_{\mathrm{B}}$ on tension $\hat{J}\doteq
JL_{\mathrm{t}}/\gamma_{\mathrm{t}}$ and torque
$\hat{\tau}\doteq\tau\phi_{\mathrm{t}}/\gamma_{\mathrm{t}}$, the two
mechanical control variables, all at constant temperature $\hat{T}\doteq
k_{\mathrm{B}}T/\gamma_{\mathrm{t}}$. The landscape in panel (a) describes how
the system contracts when the torque increases at constant tension and how the
system expands when the tension grows at constant torque.

The twist angle responds to a torque of increasing magnitude as shown in panel
(b). The response is antisymmetric and stronger at low tension than at high
tension. If we apply a torque of significant strength in positive direction,
then only twist particles with activation energies $\epsilon_{+}$ attain
significant populations.

The entropy landscape of panel (c) is more complicated. The entropy is low at
strong tension and weak torque because twist particles are mostly frozen out
owing to their high activation energies. The entropy is also low at strong
torques and low tension. Here the system is saturated with one or the other
species of twist particles, which have negative activation energies.
Elsewhere, the macrostate of the system is more strongly fluctuating, which
enhances the entropy.

Varying the scaled temperature shifts the balance between the quanta of
elastic energy carried by the particles and the ambient thermal fluctuations.
The effects are illustrated in Figs. 1(d)-(f). For all three quantities, the
slopes characterizing the landscapes become steeper as $\hat{T}$ is lowered.
Emerging are two steps for $\hat{L}$, a terrace with three levels for
$\hat{\phi}$, and two narrow ridges for $\hat{S}$.

##  IV From B-DNA to S-DNA and P-DNA

Here we generalize the previous application to describe the structural
transitions between B-DNA and two stretched conformations, underwound S-DNA
and overwound P-DNA. The default control variables are tension $J$ and torque
$\tau$. All results are convertible into the functional relations directly
probed by experiments ABLC98 ; LRS+99 ; SLCM01 ; BSG+03 ; CLH+96 ; CYL+04 .

At low torque, a gradual increase in tension is known to trigger a transition
from B-DNA to S-DNA. In Sec. V of Ref. mct1  we have already analyzed this
transition in the absence of any torsional constraints. At moderately low
tension, a gradual increase in torque is known to convert B-DNA into P-DNA. At
low tension this scenario is complicated by the formation of plectonemes, a
phenomenon investigated in Sec. VIII.

###  IV.1 Nucleation and growth of conformations

In this application we employ two host/tag pairs of level-2 nested particles
as introduced in Ref. mct1  (Sec. II.D and App. C). From the B-DNA reference
state, segments of either S-DNA or P-DNA are nucleated by the activation of
host particles and then grown by the activation of tag particles. The
combinatorial specifications of all four particle species are summarized in
Table 2.

Table 2: Capacity constants $A_{m}$ and statistical interaction coefficients
$g_{mm^{\prime}}$ for two host/tag pairs of level-2 particles.

| $m$ |  $A_{m}$
---|---|---
S-host | 1 |  $N-1$
S-tag | 2 |  $0$
P-host | 3 |  $N-1$
P-tag | 4 |  $0$

$g_{mm^{\prime}}$ |  1 |  2 | 3 |  4
---|---|---|---|---
1 | $2$ | $1$ | $1$ | $1$
2 | $-1$ | $0$ | $0$ | $0$
3 | $2$ | $1$ | $2$ | $1$
4 | $0$ | $0$ | $-1$ | $0$

The particle activation energies exhibit the standard dependence (7) on
tension and torque,

$\displaystyle\epsilon_{1}$
$\displaystyle=\gamma_{\mathrm{S}}-JL_{\mathrm{S}}-\tau\phi_{\mathrm{S}}+c_{\mathrm{S}},$
(16a) $\displaystyle\epsilon_{2}$
$\displaystyle=\gamma_{\mathrm{S}}-JL_{\mathrm{S}}-\tau\phi_{\mathrm{S}},$
(16b) $\displaystyle\epsilon_{3}$
$\displaystyle=\gamma_{P}-JL_{\mathrm{P}}-\tau\phi_{\mathrm{P}}+c_{\mathrm{P}},$
(16c) $\displaystyle\epsilon_{4}$
$\displaystyle=\gamma_{\mathrm{P}}-JL_{\mathrm{P}}-\tau\phi_{\mathrm{P}},$
(16d)

here amended with constants $c_{\mathrm{S}}$ or $c_{\mathrm{P}}$ in the two
host species, by which the cooperativity of the conformational changes is
being controlled. All specifications used in this application are inferred
from well established empirical data such as found in Refs. ABLC98 ; LRS+99 ;
SLCM01 ; CLH+96 ; CYL+04 :

$\displaystyle L_{\mathrm{S}}=L_{\mathrm{P}}\doteq
L_{\mathrm{c}}=0.24\mathrm{nm/bp},$ (17a)
$\displaystyle\phi_{\mathrm{S}}=-0.42\mathrm{rad/bp},\quad\phi_{\mathrm{P}}=+1.5\mathrm{rad/bp},$
(17b)
$\displaystyle\gamma_{\mathrm{S}}=16\mathrm{pNnm},\quad\gamma_{\mathrm{P}}=60\mathrm{pNnm}.$
(17c)

The twist angles in this application are relative to the native chirality of
B-DNA.

###  IV.2 Phase diagram

The statistical mechanical analysis proceeds as in previous applications. We
calculate the particle population densities,
$\bar{N}_{m}\doteq\lim_{N\to\infty}\langle N_{m}\rangle/N$, by solving Eqs.
(5) and (6) numerically and infer the scaled entropy, $\bar{S}(T,J,\tau)$, via
Eqs. (10). From the particle population densities $\bar{N}_{m}$,
$m=1,\ldots,4$, we infer the fraction of ds-DNA in each conformation as
follows:

$\displaystyle F_{\mathrm{S}}=\bar{N}_{1}+\bar{N}_{2},\quad
F_{\mathrm{P}}=\bar{N}_{3}+\bar{N}_{4},$ $\displaystyle
F_{\mathrm{B}}=1-F_{\mathrm{S}}-F_{\mathrm{P.}}$ (18)

Contour plots of these fractions are shown in Fig. 2(a)-(c) for low
cooperativity and in Fig. 3(a)-(c) for high cooperativity. Three regions of
predominant conformation are clearly identifiable. The crossover regions are
broad at low cooperativity. At high cooperativity, they look more like three
phase boundaries ending in one vertex.

Figure 2: Fraction of segments in the conformations of (a) S-DNA, (b) P-DNA,
and (c) B-DNA. (d) Scaled entropy. The parameter values
$c_{\mathrm{S}}=c_{\mathrm{P}}=1$ pN nm indicate low cooperativity. The
(asymptotic) phase boundaries (19) are shown as straight lines, meeting in the
triple point (20).

Figure 3: Fraction of segments in the conformations of (a) S-DNA, (b) P-DNA,
and (c) B-DNA. (d) Scaled entropy. The parameter values $c_{\mathrm{S}}=16$ pN
nm, $c_{\mathrm{P}}=15$ pN nm indicate high cooperativity. The (asymptotic)
phase boundaries (19) are shown as straight lines, meeting in the triple point
(20).

As a consistency check we have added panel (d) to Figs. 2 and 3, which shows a
contour plot of the scaled entropy, $\bar{S}/k_{\mathrm{B}}$. The level of
disorder is significantly higher in regions where two conformations mix and
even higher where all three conformations are present. The regions of mixed
conformation become narrow when cooperativity is high. Here the lines of
enhanced entropy turn into accurate markers of the three emergent phase
boundaries already identified. At the P-S border, which is realized at high
tension, little entropy is produced. However, this does not reduce the
accuracy of the entropy as a phase-boundary locator.

The B-S phase boundary at zero torque and the B-P phase boundary at very low
tension, both well-defined at high cooperativity, are consistent with
experimental benchmarks ABLC98 ; LRS+99 ; SLCM01 ; CLH+96 ; CYL+04 . The
former is also in quantitative agreement with one relevant landmark in a
different comparison of experimental data with results from our methodology,
namely in the context of torsion-free stretching SACB00 ; mct1 .

The positive slope of the B-S phase boundary is attributable to the fact that
S-DNA is underwound. Stretching at higher torque delays the B-S transition.
The negative slope of the B-P phase boundary is explained by the fact that
P-DNA is overwound. Stretching at higher torque enhances the B-P transition.
The emergent vertex at $\tau\simeq 23$ pN nm and $J\simeq 110$ pN, which is
identifiable in Fig. 3(d) with fair precision, is well established
experimentally in the form of a secondary force-extension plateau found in
torsionally constrained DNA SAB+96 ; Wang97 .

###  IV.3 Transition asymptotics

Here present the analytical form for the emergent phase boundaries, derived
from the partition function (4), which essentially informs all results in
every application. This extraction is somewhat technical, carried out for room
temperature in a high-cooperativity limit. The analysis yields three phase
boundaries in the form of straight line segments in the $(J,\tau)$-plane
meeting in one vertex:

$\tau_{\mathrm{BS}}(J)=\frac{JL_{\mathrm{c}}-\gamma_{\mathrm{S}}}{|\phi_{\mathrm{S}}|}\quad:~{}\frac{\gamma_{\mathrm{S}}}{L_{\mathrm{c}}}\leq
J\leq J_{\mathrm{T}},$ (19a)
$\tau_{\mathrm{BP}}(J)=\frac{\gamma_{\mathrm{P}}-JL_{\mathrm{c}}}{\phi_{\mathrm{P}}}\quad:~{}0\leq
J\leq J_{\mathrm{T}},$ (19b)
$\tau_{\mathrm{SP}}(J)=\tau_{\mathrm{T}}\quad:~{}J\geq J_{\mathrm{T}},$ (19c)
$J_{\mathrm{T}}=\frac{\gamma_{\mathrm{S}}\phi_{\mathrm{P}}-\gamma_{\mathrm{P}}\phi_{\mathrm{S}}}{L_{\mathrm{c}}(\phi_{\mathrm{P}}-\phi_{\mathrm{S}})}=106.8\mathrm{pN},$
(20a)
$\tau_{\mathrm{T}}=\frac{\gamma_{\mathrm{P}}-\gamma_{\mathrm{S}}}{\phi_{\mathrm{P}}-\phi_{\mathrm{S}}}=22.9\mathrm{pNnm}.$
(20b)

These phase boundaries, in accord with experimental data, are included in
Figs. 2 and 3.

The conformational fractions are governed, for the most part, by the
activation energies of the S-tags and P-tags of Table 2. This is evident in
the following leading-order asymptotic expression. In the S-phase, where
$\epsilon_{2}<0$ and $\epsilon_{4}>0$, we have

$\displaystyle F_{\mathrm{S}}$
$\displaystyle\simeq\frac{1}{2}\left(1-\frac{\sinh(\beta\epsilon_{2}/2)}{\sqrt{\sinh^{2}(\beta\epsilon_{2}/2)+e^{-\beta
c_{\mathrm{S}}}}}\right),$ (21a) $\displaystyle F_{\mathrm{P}}$
$\displaystyle=\mathrm{O}\big{(}e^{-\beta\epsilon_{3}}\big{)},$ (21b)
$\displaystyle F_{\mathrm{B}}$
$\displaystyle\simeq\frac{1}{2}\left(1+\frac{\sinh(\beta\epsilon_{2}/2)}{\sqrt{\sinh^{2}(\beta\epsilon_{2}/2)+e^{-\beta
c_{\mathrm{S}}}}}\right).$ (21c)

In the P-phase, we have $\epsilon_{2}>0$, $\epsilon_{4}<0$, yielding

$\displaystyle F_{\mathrm{S}}$
$\displaystyle=\mathrm{O}\big{(}e^{-\beta\epsilon_{1}}\big{)}$ (22a)
$\displaystyle F_{\mathrm{P}}$
$\displaystyle\simeq\frac{1}{2}\left(1-\frac{\sinh(\beta\epsilon_{4}/2)}{\sqrt{\sinh^{2}(\beta\epsilon_{4}/2)+e^{-\beta
c_{\mathrm{P}}}}}\right),$ (22b) $\displaystyle F_{\mathrm{B}}$
$\displaystyle\simeq\frac{1}{2}\left(1+\frac{\sinh(\beta\epsilon_{4}/2)}{\sqrt{\sinh^{2}(\beta\epsilon_{4}/2)+e^{-\beta
c_{\mathrm{P}}}}}\right).$ (22c)

Positive activation energies, $\epsilon_{2}>0$ and $\epsilon_{4}>0$, suppress
S-tags and P-tags in the B-phase:

$\displaystyle F_{\mathrm{S}}$ $\displaystyle\simeq
e^{-\beta\epsilon_{2}}e^{-\beta c_{\mathrm{S}}},$ (23a) $\displaystyle
F_{\mathrm{P}}$ $\displaystyle\simeq e^{-\beta\epsilon_{4}}e^{-\beta
c_{\mathrm{P}}},$ (23b) $\displaystyle F_{\mathrm{B}}$ $\displaystyle\simeq
1-e^{-\beta\epsilon_{2}}e^{-\beta
c_{\mathrm{S}}}-e^{-\beta\epsilon_{4}}e^{-\beta c_{\mathrm{P}}}.$ (23c)

The results (21b) and (22a) mean that those fractions are exponentially
suppressed throughout that particular phase. These functional forms in
combination with specifications (17) are testable predictions.

##  V Supercoils

We begin our analysis of supercoils by returning to the idealized molecular
chain with a ladder-like reference state introduced in Sec. III. We now
consider the possibility of two different responses to torque: the twisting of
bonds or monomers and the formation of supercoils. We distinguish two
scenarios, one pertaining to high tension and the other to low tension.

The first scenario (Sec. VI) describes a system with low torsional stiffness.
When a torque of increasing strength is applied under significant (constant)
tension, the system first responds via twist contraction as described in Sec.
III. When the twist response approaches saturation, the system is in need of a
new way to respond to yet stronger torque. That mode involves the formation of
supercoils, which in this case are loops of highly twisted chain (as in a
rope).

The second scenario (Sec. VII) describes a system with high torsional
stiffness. At low (constant) tension, the primary response to an applied
torque of increasing strength is the formation of supercoils, which in this
case are loops of largely untwisted chain (as in a garden hose). When we
increase the tension under torsional constraint, supercoiled chain is
gradually converted into twisted chain. With some modifications, either
scenario is adaptable to DNA applications. One realization of a buckling
transition is demonstrated in Sec. VIII.

The modeling of supercoil conformations employs level-2 nested particles as
introduced in App. C of mct1 . Chain segments of left-handed or right-handed
twist are activated by twist particles and segments of left-handed or right-
handed writhe by supercoil particles. We use four species (a host, a hybrid,
and two tags) for each sense of chirality. Symbolic representations of the
eight species of particles are shown in Fig. 4.

Figure 4: Four species of level-2 twist particles (hosts 1, 3 and tags 2, 4)
and four species of supercoil particles (hybrids 5, 7 and tags 6, 8)
constituting two nested structures (one for each sense of chirality). Positive
(negative) torque lowers the activation energies of particles 1, 2, 5, 6 (3,
4, 7, 8).

Hosts 1 and 3 nucleate segments of twisted chain with opposite chirality out
of the untwisted ladder reference state. Tags 2 and 4 contribute to the growth
of twisted-chain segments. Hybrids 5 and 7 nucleate segments of supercoil. The
growth of supercoil segments is governed by tags 6 and 8. In the first
scenario (Sec. VI), the supercoil segments grow out of twisted chain (e.g. by
tags 6 replacing tags 2), whereas in the second scenario (Sec. VII) they grow
out of untwisted chain (e.g. by tags 6 replacing elements of reference state).
The level-2 particles used here offer two key advantages over the level-1
particles used in Sec. III. They provide a parameter which controls the
cooperativity of twisted chain segments and they facilitate the model
extension to include supercoils.

The combinatorial analysis as previously described in different contexts
LVP+08 ; copic ; picnnn ; pichs ; cohetra  yields the specifications compiled
in Table 3. The dependence on tension and torque of the particle activation
energies has the standard form (7), here again rewritten with a change of sign
for convenience:

$\epsilon_{m}=\gamma_{m}+JL_{m}-\tau\phi_{m}.$ (24)

Table 3: Statistical interaction coeffficients $g_{mm^{\prime}}$ of the four species of twist (tw) particles and the four species of supercoil (sc) particles. The capacity constants are $A_{1}=A_{3}=N-2$ and $A_{m}=0$, $m=2,4,5,6,7,8$. $g_{mm^{\prime}}$ | $m\big{\backslash}m^{\prime}$ | $1$ | $~{}~{}2$ | $3$ | $~{}~{}4$ | $~{}~{}5$ | $~{}~{}6$ | $~{}~{}7$ | $~{}~{}8$
---|---|---|---|---|---|---|---|---|---
tw host | $1$ | $2$ | $1$ | $2$ | $1$ | $2$ | $1$ | $2$ | $1$
tw tag | $2$ | $-1$ | $0$ | $0$ | $0$ | $-1$ | $0$ | $0$ | $0$
tw host | $3$ | $1$ | $1$ | $2$ | $1$ | $2$ | $1$ | $2$ | $1$
tw tag | $4$ | $0$ | $0$ | $-1$ | $0$ | $0$ | $0$ | $-1$ | $0$
sc hybrid | $5$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
sc tag | $6$ | $0$ | $0$ | $0$ | $0$ | $-1$ | $0$ | $0$ | $0$
sc hybrid | $7$ | $0$ | $0$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
sc tag | $8$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-1$ | $0$

The two kinds of nucleation and growth processes are controlled by the
parameters $\gamma_{m},L_{m},\phi_{m}$. We set

$\displaystyle\gamma_{m}>0,\quad L_{m}>0\quad:~{}m=1,\ldots,8,$
$\displaystyle\phi_{m}>0\quad:~{}m=1,2,5,6,$ (25)
$\displaystyle\phi_{m}<0\quad:~{}m=3,4,7,8.$

Symmetry considerations require the following relations between parameters:

$\displaystyle\gamma_{1}=\gamma_{3},\quad\gamma_{2}=\gamma_{4},\quad\gamma_{5}=\gamma_{7},\quad\gamma_{6}=\gamma_{8},$
$\displaystyle L_{1}=L_{3},\quad L_{2}=L_{4},\quad L_{5}=L_{7},\quad
L_{6}=L_{8},$ (26)
$\displaystyle\phi_{1}=-\phi_{3},\quad\phi_{2}=-\phi_{4},\quad\phi_{5}=-\phi_{7},\quad\phi_{6}=-\phi_{8}.$

In applications to chains with intrinsic chirality these relations need to be
modified as will be explored in a separate study mct3 .

##  VI Supercoils at high tension

This first scenario is designed (by parameter setting) to produce supercoil
segments made of highly twisted chain. Under increasing torque, supercoil
particles crowd out twist particles. The former thus incorporate two kinds of
contraction lengths and two kinds of angles, namely twist and writhe. This is
accommodated by specifications which satisfy the inequalities,

$\displaystyle L_{5}>L_{1},\quad L_{7}>L_{3},\quad L_{6}>L_{2},\quad
L_{8}>L_{4},$ (27a)
$\displaystyle\phi_{5}>\phi_{1},\quad|\phi_{7}|>|\phi_{3}|,\quad\phi_{6}>\phi_{2},\quad|\phi_{8}|>|\phi_{4}|.$
(27b) The additional inequalities,
$\gamma_{5}>\gamma_{1},\quad\gamma_{7}>\gamma_{2},\quad\gamma_{6}>\gamma_{2},\quad\gamma_{8}>\gamma_{4},$
(27c)

facilitate the empirical requirement that supercoiling is a phenomenon
realized in a highly twisted chain.

###  VI.1 Low torque: twisted chain

With the parameter setting satisfying the constraints (27) there exists a
regime of low torque where only twist particles are activated in significant
numbers. Here supercoil particles have high activation energies and are
effectively frozen out.

In the statistical mechanical analysis, the limit of infinitely high
activation energies for supercoil particles implies that

$w_{m}\to\infty\quad:~{}m=5,\ldots,8,$ (28)

which, in turn, has the consequence that the linear Eqs. (6) yield vanishing
population densities for supercoil particles:

$\bar{N}_{m}\doteq\frac{\langle N_{m}\rangle}{N}\to 0\quad:~{}m=5,\ldots,8.$
(29)

The polynomial Eqs. (5) for the remaining variables, $w_{m}$, $m=1,\ldots,4$,
are then of lower order, cubic in this case. The (unique) physical solution
determines the population densities of twist particles via Eqs. (6) for
$m=1,\ldots,4$ and the Gibbs free energy $\bar{G}(T,J,\tau)$ via (8). The
contraction length $\bar{L}$, the entropy $\bar{S}$, and the twist angle
$\bar{\phi}$ are calculated from from $\bar{G}(T,J,\tau)$ via derivatives as
in (9) or via population densities via (10) and (11) properly rescaled.

Figure 5: Population densities of (a) host particles and (b) tag particles,
both versus tension $\hat{J}$ and torque $\hat{\tau}$ at constant temperature
$\hat{T}=1$ and zero cooperativity, $\Delta\gamma_{\mathrm{t}}=0$. The nine
contour lines from dark to bright in each panel are at (a)
$0.025,\ldots,0.225$, and (b) $0.096,\ldots,0.864$. Panels (c)-(g) show the
same two quantities plus contraction length $\hat{L}$, twist $\hat{\phi}$, and
entropy $\hat{S}$ plotted versus $\hat{\tau}$ at constant tension $\hat{J}=1$
and temperature $\hat{T}=1$ for cooperativity with strength increasing from
zero: $\Delta\gamma_{\mathrm{t}}/\gamma_{\mathrm{t}}=0,2,4$. Panel (h) shows
contraction length versus twist angle.

In Fig. 5 we present some explicit results. The specifications in use are

$\displaystyle L_{1}=L_{2}=L_{3}\doteq L_{\mathrm{t}},$
$\displaystyle\phi_{1}=\phi_{2}=-\phi_{3}=-\phi_{4}\doteq\phi_{\mathrm{t}},$
$\displaystyle\gamma_{2}=\gamma_{4}\doteq\gamma_{\mathrm{t}},\quad\gamma_{1}=\gamma_{3}=\gamma_{\mathrm{t}}+\Delta\gamma_{\mathrm{t}}.$
(30)

For the graphical presentation we adopt the following rescaling conventions:

$\displaystyle\hat{L}\doteq\frac{|\bar{L}|}{L_{\mathrm{t}}},\quad\hat{\phi}\doteq\frac{\bar{\phi}}{\phi_{\mathrm{t}}},\quad\hat{S}\doteq\frac{\bar{S}}{k_{\mathrm{B}}},$
$\displaystyle\hat{J}\doteq\frac{JL_{\mathrm{t}}}{\gamma_{\mathrm{t}}},\quad\hat{\tau}\doteq\frac{\tau\phi_{\mathrm{t}}}{\gamma_{\mathrm{t}}},\quad\hat{T}\doteq\frac{k_{\mathrm{B}}T}{\gamma_{\mathrm{t}}}.$
(31)

In panels (a) and (b) we show the population densities of hosts 1, 3 and tags
2, 4 versus torque and tension for the case of zero cooperativity and fixed
temperature. Panels (c)-(g) show the same population densities plus the
thermodynamic functions $\hat{L}$, $\hat{\phi}$, $\hat{S}$, all versus torque
at fixed tension, temperature (one value each), and cooperativity (three
values). The dependence of contraction length on twist angle is shown in panel
(h).

At fixed tension and with positive torque of increasing strength, twisted
segments are being nucleated through the activation of hosts 1. Twisted
segments grow in length through the activation of tags 2. Adjacent segments of
twisted chain merge as hosts are crowded out by tags. Negative torque favors
the activation and growth of hosts 3 and tags 4. Any increase in the
cooperativity is controlled by raising the nucleation threshold
$\Delta\gamma_{\mathrm{t}}$ from zero. Fewer extended segments are being
nucleated. However, once they are nucleated, they grow more rapidly with
torque of increasing strength. The changes in contraction length and twist
angle take place more abruptly. The entropy is lower overall and peaks when
the torque crosses the nucleation threshold.

Note the similarities and differences between temperature effects in Fig.
1(d)-(f) and Fig. 5(e)-(g). For the quantitative analysis of transformations
between conformations in DNA under tension and torque, effects of
cooperativity are important. It takes nested particles to model cooperativity
in an natural way. The dependence of contraction length $\hat{L}$ on
$\hat{\phi}$ as depicted in Fig. 5(h) is a key quantity, directly accessible
to experiments on DNA. Here we see what twist alone produces. This simple,
near linear dependence will be modified by the presence of supercoiling in
some settings.

###  VI.2 High torque: twisted and supercoiled chain

At sufficiently strong (positive) torque, the activation energies of supercoil
particles 5,6 descend below those of twist particles 1,2 as illustrated in the
inset to Fig. 6 for one representative case with the following specifications:

$\displaystyle\gamma_{1}=\gamma_{3}\doteq\gamma_{\mathrm{t}}+\Delta\gamma_{\mathrm{t}},\quad\gamma_{2}=\gamma_{4}\doteq\gamma_{\mathrm{t}},$
$\displaystyle\gamma_{5}=\gamma_{7}\doteq\gamma_{\mathrm{s}}+\Delta\gamma_{\mathrm{s}},\quad\gamma_{6}=\gamma_{8}\doteq\gamma_{\mathrm{s}},$
$\displaystyle L_{1}=L_{2}=L_{3}=L_{4}\doteq L_{\mathrm{t}},\quad
L_{5}=L_{6}=L_{7}=L_{8}\doteq L_{\mathrm{s}},$
$\displaystyle\phi_{1}=\phi_{2}=-\phi_{3}=-\phi_{4}\doteq\phi_{\mathrm{t}},$
$\displaystyle\phi_{5}=\phi_{6}=-\phi_{7}=-\phi_{8}\doteq\phi_{\mathrm{s}},$
(32)
$\displaystyle\beta\gamma_{\mathrm{t}}=1,\quad\gamma_{\mathrm{s}}/\gamma_{\mathrm{t}}=2.5,\quad\Delta\gamma_{\mathrm{t}}/\gamma_{\mathrm{t}}=0.3,\quad\Delta\gamma_{\mathrm{s}}/\gamma_{\mathrm{t}}=0.3,$
$\displaystyle JL_{\mathrm{t}}/\gamma_{\mathrm{t}}=0.3,\quad
JL_{\mathrm{s}}/\gamma_{\mathrm{t}}=0.45,\quad\phi_{\mathrm{s}}/\phi_{\mathrm{t}}=1.5.$
(33)

At constant tension, the $\epsilon_{m}$ vary linearly with torque. Positive
(negative) torque lowers $\epsilon_{m}$ for $m=1,2,5,6$ ($m=3,4,7,8$). By
design, the slope is steeper and the intercept higher for supercoil particles,
$m=5,6,7,8$, than for twist particles, $m=1,2,3,4$. Cooperativity for twist
(supercoil) particles displaces the line for hosts 1,3 (hybrids 5,7) upward
relative to the line for tags 2,4 (6,8).

Figure 6: Population densities of twist particles 1, 3 (hosts), 2, 4 (tags),
and supercoil particles 5, 7 (hybrids), 6, 8 (tags) versus torque $\hat{\tau}$
at constant tension $\hat{J}$ and temperature $\hat{T}$ with specifications
(VI.2), (VI.2). The inset shows the activation energies $\beta\epsilon_{m}$
versus torque $\hat{\tau}$.

The statistical mechanical analysis of the general case with all eight
particles in play now requires that we solve the nonlinear Eqs. (5) and the
linear Eqs. (6) for all eight species. The former reduce to a fifth-order
polynomial equation with a unique physical solution. In the main plot of Fig.
6 we show the population density $\bar{N}_{m}$ for each particle species
versus torque at fixed tension and temperature. It pertains to a tension of
moderate strength and a response of moderate cooperativity for both twist and
supercoil particles. The temperature selected is also of intermediate value.

At zero torque only hosts 1 and 3 have significant populations. They represent
very short segments of twisted chain, activated by thermal fluctuations with
no bias in orientation. A weak positive torque enhances such fluctuations of
one orientation (hosts 1) at the expense of the other (hosts 3). With
increasing (positive) torque the twist segments nucleated by hosts 1 grow in
size through the activation of an increasing number of tags 2. As the segments
of twisted chain grow in size they begin to merge, which reduces their number.
This process is reflected in a decrease of the density of hosts,
$\bar{N}_{1}$, as the density of tags, $\bar{N}_{2}$, continues to increase.

The threshold for the appearance of supercoil particles is reached when their
activation energies descend below those of the twist particles (see inset).
The sharpness of this threshold depends on cooperativity. At this threshold,
supercoil segments begin to nucleate. The mechanism of this nucleation is the
activation of hybrids 5 and the mechanism for the growth of supercoil segments
is the activation of tags 6. Under yet stronger torque the supercoil particles
begin to crowd out twist particles, which means that the fraction of
supercoiled chain increases at the expense of merely twisted chain. As the
supercoil segments grow, they begin to merge just as the twisted segments did
at weaker torque. This is reflected in the decrease of $\bar{N}_{5}$ (hybrids)
and increase of $\bar{N}_{6}$ (tags). At very strong torque, the system is in
the conformation of a single supercoil segment, composed of one host 1, one
hybrid 5, and a macroscopic number of tags 6.

Figure 7: (a) Contraction distance $\hat{L}$, (b) linkage $\hat{\phi}$, and
(c) entropy $\hat{S}$, all versus torque $\hat{\tau}$ at constant tension
$\hat{J}$ and temperature $\hat{T}$. Panel (d) shows $\hat{L}$ versus
$\hat{\phi}$. The three curves in (c) are for $\hat{T}=1,0.5,0.25$. Data for
intermediate $\hat{T}$ are not shown in (a) and (b). The remaining
specifications are from (VI.2), (VI.2).

The effects of supercoil formation on length contraction, linkage angle, and
entropy are shown in Fig. 7. The characteristic features associated with
supercoils stand out in a comparison of data at two or three different
temperatures. These features are especially prominent when Figs. 7(a)-(c) are
compared with Figs. 5(e)-(g), where the supercoil response is suppressed. From
Fig. 6 we know that the activation of supercoils sets in at $\hat{\tau}\gtrsim
3.5$. The signature of supercoil formation is an enhancement in linkage at the
cost of an augmented contraction. Both effects are barely visible at the
highest $\hat{T}$ but very clearly at the lowest $\hat{T}$.

The entropy curve at the highest $\hat{T}$ has a broad maximum for
intermediate torque. This reflects strong fluctuations involving competing
conformations. All of them are represented by particles with activation
energies near zero. Only at high torque does the supercoil conformation become
predominant and particles 6 crowd out all others, which is reflected in a
pronounced entropy drop. At the intermediate temperature, the signature of
supercoil formation becomes discernible in the entropy data as a shoulder-like
feature at $\hat{\tau}\simeq 3.5$. The sharpened maximum (at $\hat{\tau}\simeq
1.5$) marks the torque threshold for the activation of twisted segments. The
features associated with twisted chain are similar to what we have already
described in the context of Fig. 5. The appearance of supercoils manifests
itself by way of sharp increases in contraction length and linkage as well as
by a sharp spike in the entropy.

###  VI.3 Showcase for particle nesting

In conclusion of Sec. VI with focus on high-tension supercoils and in
anticipation of Sec. VII, where our focus will shift to low-tension
supercoils, we are well positioned to highlight the role of particle nesting
within the framework of statistically interacting particles. Particle nesting
is the key feature in the statistical mechanical analysis of this rather
complex behavioral trait manifest in stretched and twisted polymeric chains.

The same set of eight particle species, introduced in Sec. V and consisting of
two hierarchies, one for each sense of chirality, can be used to describe, in
flexible polymers, the appearance of supercoils out of highly twisted chains
and, with different parameter settings for stiff polymers, the formation of
supercoils out of largely untwisted chains. The nesting structure, which is
determined by the combinatorial specifications given in Table 3, is identical
for low-tension and high-tension supercoils even though some particle species
take on very different roles in the two applications. The specific role is
determined by energetic parameters.

The same two hierarchies thus describe, in one parameter setting, how a
twisted chain converts into a supercoiled chain under increasing torque at
fixed tension (shown earlier in Sec. VI) and, in a different parameter
setting, how supercoiled chain gives way to twisted chain when torque is
increased at fixed tension (to be demonstrated in Sec. VII). With some
modifications this model has applications for ds-DNA, which we plan to explore
in continuation of this project mct3 .

##  VII Supercoils at low tension

In molecular chains of strong torsional stiffness, supercoils tend to form out
of segments that are weakly twisted at most, within a regime of low tension
and not too high torque. It then suffices that we reverse the inequalities
(27b) and (27c) into

$\displaystyle\phi_{5}<\phi_{1},\quad|\phi_{7}|<|\phi_{3}|,\quad\phi_{6}<\phi_{2},\quad|\phi_{8}|<|\phi_{4}|,$
$\displaystyle\gamma_{5}<\gamma_{1},\quad\gamma_{7}<\gamma_{2},\quad\gamma_{6}<\gamma_{2},\quad\gamma_{8}<\gamma_{4},$
(34)

while leaving inequalities (27a) intact. In consequence, the dependence on
torque of the activation energies for twist particles and supercoil particles
is interchanged (compare insets to Figs. 8 and 6). We change four of the six
scaling conventions (VI.1) for this section as follows:

$\displaystyle\hat{\phi}\doteq\frac{\bar{\phi}}{\phi_{\mathrm{s}}},\quad\hat{J}\doteq\frac{JL_{\mathrm{t}}}{\gamma_{\mathrm{s}}},\quad\hat{\tau}\doteq\frac{\tau\phi_{\mathrm{t}}}{\gamma_{\mathrm{s}}},\quad\hat{T}\doteq\frac{k_{\mathrm{B}}T}{\gamma_{\mathrm{s}}}.$
(35)

The energetic specifications for the case analyzed in the following are (VI.2)
and

$\displaystyle\beta\gamma_{\mathrm{s}}=1,\quad\gamma_{\mathrm{t}}/\gamma_{\mathrm{s}}=3.0,\quad\Delta\gamma_{\mathrm{t}}/\gamma_{\mathrm{s}}=0.3,\quad\Delta\gamma_{\mathrm{s}}/\gamma_{\mathrm{s}}=0.3,$
$\displaystyle JL_{\mathrm{t}}/\gamma_{\mathrm{s}}=0.1,\quad
JL_{\mathrm{s}}/\gamma_{\mathrm{s}}=0.15,\quad\phi_{\mathrm{t}}/\phi_{\mathrm{s}}=1.5.$
(36)

Figure 8: Population densities of twist particles 1, 3 (hosts), 2, 4 (tags),
and supercoil particles 5, 7 (hybrids), 6, 8 (tags) versus torque $\hat{\tau}$
at constant tension $\hat{J}$ and temperature $\hat{T}$. The inset shows the
activation energies $\beta\epsilon_{m}$ versus torque $\hat{\tau}$ for the
case with specifications (VII).

At zero torque, hosts 1 and 3, which nucleate segments of twisted chain, still
have the highest populations of all particles, now greatly reduced. The next
highest population densities pertain to hybrids 5 and 7. Unlike in Sec. VI,
weak positive (negative) torque does not grow twisted segments by activating
tags 2 (tags 4). Instead it nucleates supercoil segments via the activation of
hybrids 5 (hybrids 7) and grows with tags 6 (tags 8).

As the torque continues to gain strength, the supercoiled segments begin to
merge. The signature of that effect is a decrease in the populations of
hybrids 5, (hybrids 7) combined with an increase in the populations of tags 6
(tags 8). Only at yet stronger torque is the torsional stiffness of the chain
overcome. Supercoiled segments now convert into segments of twisted chain of
increasing size. The population of tags 6 (tags 8) collapses while the
population of tags 2 (tags 4) shoots up.

Figure 9: (a) Contraction distance $\hat{L}$, (b) linkage $\hat{\phi}$, and
(c) entropy $\hat{S}$, all versus torque $\hat{\tau}$ at constant tension
$\hat{J}$ and temperature $\hat{T}$. Panel (d) shows $\hat{L}$ versus
$\hat{\phi}$. The three curves in (c) are for $\hat{T}=1,0.5,0.25$. Data for
the intermediate temperature are not shown in (a), (b), and (d). The remaining
specifications are from (VI.2), (VII).

The effects on length contraction, linkage, and entropy of the formation of
low-tension supercoils at weak torque and their conversion into twisted chain
at stronger torque are shown in Fig. 9. Some of the contrasts to the
corresponding plots in Fig. 7 are striking. In each panel we can discern the
following three major processes: the nucleation of supercoil segments at
$|\hat{\tau}|\simeq 1$, the merging of supercoil segments at
$2\lesssim|\hat{\tau}|\lesssim 3.5$, and their conversion into long twisted
segments at $|\hat{\tau}|\simeq 4$.

The nucleation of supercoil segments is gradual at the highest temperature and
becomes rather precipitous at the lowest temperature, graphically represented,
respectively, by soft and sharp increases in length contraction, linkage, and
entropy. The merging of supercoil segments is associated with small increases
in length contraction and linkage, but a marked decrease in entropy,
especially at the lower temperatures. The conversion of supercoiled chain into
twisted chain combines an increase in linkage with a decrease in contraction
length, which is gradual at the highest $T$ and abrupt at the lowest $T$. The
effect on entropy is a spike, which is small but prominent at low $T$. With
increasing $T$, this signal gets more and more washed out by thermal
fluctuations into a shoulder-like feature.

##  VIII Buckling transition

One intensively investigated phenomenon in single-molecule experiments on ds-
DNA under tension and torque is the buckling transition, specifically the
rather abrupt formation of plectonemes when the twist angle is gradually
advanced under constant low tension SAB+96 ; Mark98 ; ABLC98 ; SABC98 ; HYZ99
; SAB+99 ; BM00 ; ZZY00 ; SABC00 ; SACB00 ; BBS03 ; Neuk04 ; Mark07 ; FDS+08 ;
DFS+09 ; SW09 ; MN13 ; ELS13 . Here we demonstrate that the system of eight
species of statistically interacting particles portrayed in Secs. VI does
indeed include a parameter regime that features a buckling transition.

We return to the scaling convention (VI.1) with a single modification,

$\displaystyle\hat{L}\doteq\frac{|\bar{L}|}{L_{\mathrm{s}}}.$ (37)

We leave the specifications (VI.2) unchanged and modify the specifications
(VI.2) to

$\displaystyle\beta\gamma_{\mathrm{t}}=1,\quad\gamma_{\mathrm{s}}/\gamma_{\mathrm{t}}=2.5,\quad\Delta\gamma_{\mathrm{t}}/\gamma_{\mathrm{t}}=0.3,\quad\Delta\gamma_{\mathrm{s}}/\gamma_{\mathrm{t}}=0.3,$
$\displaystyle JL_{\mathrm{t}}/\gamma_{\mathrm{t}}=0.03,\quad
JL_{\mathrm{s}}/\gamma_{\mathrm{t}}=0.3,\quad\phi_{\mathrm{s}}/\phi_{\mathrm{t}}=1.5.$
(38)

These modifications have little impact on the population densities found for
the case analyzed in Sec. VI.2 and shown in Fig. 6. The largest difference
(not shown) is an enhancement in the population of twist tags 2 and 4.

In our methodology, particles from the same species with similar activation
densities (implying similar population densities) can describe very different
physical phenomena. The differences are encoded in the specifications (VIII)
and visually enhanced in the modified scaling convention (37). The new
physics, illustrated in Fig. 10, includes the buckling transition, of which
there is no hint in Fig. 7.

Figure 10: (a) Contraction distance $\hat{L}$, (b) twist angle $\hat{\phi}$,
and (c) entropy $\hat{S}$, all versus torque $\hat{\tau}$ at constant tension
$\hat{J}$ and temperature $\hat{T}$. Panel (d) shows $\hat{L}$ versus
$\hat{\phi}$, demonstrating the buckling transition. The two curves in each
panel are for $\hat{T}=1,0.5$. The remaining specifications are from (VI.2),
(VI.2).

All it took was that we assigned a smaller contraction length to twist
particles. This change is conspicuous in the panels (a) and (b) of the two
graphs, less so in panel (c). The buckling transition is manifest in panel (d)
in the form of a sharp increase in the rate of contraction. As is typical of
phase behavior in one-dimensional systems, what looks like real transitions
are, in fact, crossovers whose width varies with scaled temperature.

The energetics of the relevant particles in the case under scrutiny here is
such that fluctuations are sufficiently strong at $\hat{T}=1.0$ to remove any
sign of a transition. Yet, at $\hat{T}=0.5$, the abrupt initiation of
supercoiling is clearly visible in the sharply edged curve of contraction
length versus twist angle. Keep in mind that a change in $\hat{T}$ does not
have to mean a change in temperature.

We should like to emphasize that this scenario is not meant to be a model for
the buckling transition as observed in ds-DNA. Nor is it meant to claim
predictive power of our methodology in regard to this phenomenon. What will be
necessary to achieve these goals is to fix model parameters with empirical
evidence drawn from experimental data unrelated to the buckling transition and
then test the supercoil modeling introduced here, appropriately generalized to
include native chirality, for its predictive power regarding the buckling
transition. This strategy worked well in Sec. V of Ref. mct1 ) for the force
extension characteristics of torsionally unconstrained DNA stretching and will
be adapted in the continuation of this project mct3  to interpret the
ubiquitous “hat curves” found experimentally in evidence of overwound and
underwound DNA buckling SAB+96 ; Mark98 ; ABLC98 ; SABC98 ; HYZ99 ; SAB+99 ;
BM00 ; ZZY00 ; SABC00 ; SACB00 ; BBS03 ; Neuk04 ; Mark07 ; FDS+08 ; DFS+09 ;
SW09 ; MN13 ; ELS13 .

##  IX Conclusion and outlook

In this work we have extended a method of statistical mechanical analysis
based on statistically interacting quasi-particles Hald91a ; Wu94 ; Isak94 ;
Anghel ; NA14 ; LVP+08 ; copic ; picnnn ; pichs ; LMK09 ; PMK07 ; sivp ,
previously adapted to molecular molecular chains under tension mct1 . The
extension includes torsional constraints and adds torque to tension as a
second mechanical control variable. All thermodynamic relations describing
quasistatic processes involving combinations of stretching or contracting,
twisting and supercoiling are derived from a single partition function.

We have introduced the extended method of analysis in applications to
idealized ladder systems and then demonstrated its usefulness for two
applications to DNA: (i) the conversion of (native) B-DNA into (underwound)
S-DNA or (overwound) P-DNA when subjected to controlled variations of tension
and torque, (ii) the conversion between twist chirality and plectonemic
chirality when the molecular chain is subjected to controlled variations of
linkage and extension. An arena for applications to the buckling of overwound
and underwound ds-DNA has thus been opened up and is one natural extension of
this project which we intend to explore mct3 .

A second projected extension has its focus on the inclusion of interactions
between molecular chains and molecules of the embedding fluid (beyond its role
as a heat bath). This extension addresses the effects of intercalation on
force-extension characteristics, the role of melting bubbles, and the observed
manifestations of hysteresis PQSC94 ; RB01 ; WWRB02 ; DMZ+02 ; WR02 ; VMRW05 ;
PHMV05 ; SMRW08 ; WPG08 ; FCMY10 ; Zoli11 ; ZCF+12 ; BEB+12 ; ZCL+13 ; BMLB14
; BEB+14 ; APRW16 . Some of these effects can be described within the same
framework of statistical mechanical analysis as as anticipated in Sec. I of
Ref. mct1 . However, an extension of this methodology to include kinetics will
be necessary and promises to significantly widen its scope mct3 .

## References

  * (1) J. F. Marko and E. D. Siggia, Fluctuations and supercoiling of DNA, Science 265, 508 (1994).
  * (2) J. F. Marko and E. D. Siggia, Statistical mechanics of supercoiled DNA, Phys. Rev. E 52, 2912 (1995).
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# Stress inside the pion in holographic light-front QCD

Yang Li  University of Science and Technology of China, Hefei, Anhui 230026,
China  James P. Vary  Iowa State University, Ames, IA 50011, USA

###### Abstract

In this work, we propose a method to compute the gravitational form factor
$D(Q^{2})$ in holographic QCD by exploiting the remarkable correspondence
between semi-classical light-front QCD and semi-classical field theories in
wrapped spacetime in 5D. The use of light-front holography bridges physics at
large $Q^{2}$ as attained in light-front QCD and physics at small $Q^{2}$
where the coupling to scalar and tensor fields, i.e. glueballs, are dominant.
As an application, we compute the $D$-term for the pion, and compare the
results with recent lattice simulations.

## Introduction

The hadronic energy-momentum tensor (EMT) characterizes the energy, momentum
and stress distributions inside the composite systems of quarks held together
by the strong force. It also controls how the hadronic matter – 99% of all
visible matter – gravitates. Due to the strong-coupling nature of QCD, a
thorough understanding of the hadronic EMT is still missing [1]. Indeed, one
of the global charges associated with the hadronic EMT, the $D$-term, is
dubbed as the “last global unknown” [2]. Unlike the mass and spin, this term
is not constrained by global conservations laws. It encodes dynamical
information of the system, such as the force balance and mechanical stability.
The $D$-term has recently attracted attention from both the theory and
experimental communities [3].

Light-front QCD (LFQCD) provides a crucial nonperturbative picture of the
hadronic EMT. The gravitational form factors (GFFs) are related to the second
Mellin moments of the generalized parton distributions (GPDs), which also
allows a further decomposition of the total energy, spin and stress
distributions into quark, gluon and anomalous contributions [4, 5]. The light-
front wave function (LFWF) representation also affords a direct microscopic
interpretation of the hadronic EMT [6, 7]. The LFWFs can be obtained by
diagonalizing the QCD Hamiltonian quantized at fixed light-front time
$x^{+}=x^{0}+x^{3}$ as demonstrated by the basis light-front quantization
approach (BLFQ [8]). Recent strides in this direction include successful
description of quarkonium spectra [9, 10, 11, 12, 13, 14], radiative
transitions [15, 16], form factors [17, 18], nucleon valence, sea and gluon
parton distributions (PDFs, [20, 19, 21, 23, 22, 24, 25]), and transverse
momentum distributions [26, 27, 28].

One of the most remarkable features of light-front QCD is its connection to
the holographic view of QCD [29] (see Ref. [30] for a review). Holographic QCD
or AdS/QCD is inspired by the AdS/CFT correspondence, a duality relating
strongly coupled gauge theory with weakly interacting gravity theory, and
paves the way to access the strongly coupled regime of QCD while retaining the
analytic power [31, 32, 33, 34].

One of the leading examples is soft-wall AdS/QCD [35]. This model introduces a
scale through a dilaton field $\phi(z)=\kappa^{2}z^{2}$ to break the conformal
symmetry to match QCD. The scale parameter $\kappa=0.388\,\mathrm{GeV}$ is
obtained by fitting to the $\rho$ mass. The hadron mass spectra successfully
reproduce the Regge trajectory, $M^{2}_{n}=\alpha n+\beta$. Dynamical
information of the hadrons such as the decay constant, electromagnetic form
factors and transition form factors was also computed and compared with
experiments with reasonable agreement [36, 37, 38, 39, 40, 41, 42, 43, 45, 44,
46]. The fifth dimension $z$ of the AdS space in holographic QCD is shown to
be associated with the parton impact parameter
$\zeta_{\perp}=\sqrt{x(1-x)}r_{\perp}$, where $x$ is the longitudinal momentum
fraction of the parton and $r_{\perp}$ is the transverse separation of the
partons. This striking correspondence bridges the holographic view with the
partonic picture and provides an avenue to compute PDFs as well as GPDs in
holographic QCD [47, 48, 49, 50].

In this note, we compute the GFF $D_{\pi}(Q^{2})$ of the pion in holographic
light-front QCD (HLFQCD). This form factor emerges from the covariant
decomposition of the pionic matrix element of the EMT operator [51],

$\langle p+q|T^{\mu\nu}(0)|p\rangle\\\
=2P^{\mu}P^{\mu}A_{\pi}(-q^{2})+\frac{1}{2}(q^{\mu}q^{\nu}-q^{2}g^{\mu\nu})D_{\pi}(-q^{2})$
(1)

where $q=p^{\prime}-p,P=(p^{\prime}+p)/2$. The GFFs $A_{\pi}$ and $D_{\pi}$
can be obtained from two EMT components within the Drell-Yan frame
$q^{+}=0,P_{\perp}=0$,

$\displaystyle\langle p+q|T^{++}|p\rangle=$ $\displaystyle
2P^{+}P^{+}A_{\pi}(Q^{2}),$ (2) $\displaystyle\langle p+q|T^{+-}|p\rangle=$
$\displaystyle(2M^{2}_{\pi}+{\frac{1}{2}}Q^{2})A_{\pi}(Q^{2})+Q^{2}D_{\pi}(Q^{2})$
(3)

Here we adopt light-front coordinates $v^{\pm}=v^{0}\pm v^{3}$ and
$\vec{v}_{\perp}=(v^{1},v^{2})$ for an 4-vector $v^{\mu}$ [1, 52], and
$Q^{2}=-q^{2}=q_{\perp}^{2}$. Since $x^{+}$ in light-front dynamics is the
“time”, $T^{+\mu}$ is the conserved Noether current associated with the
conserved 4-momentum,

$P^{\mu}=\frac{1}{2}\int{\mathrm{d}}x^{-}{\mathrm{d}}^{2}x_{\perp}\,T^{+\mu}(x).$
(4)

The conservation of 4-momentum implies

$A_{\pi}(0)=1,\quad\lim_{Q\to 0}Q^{2}D_{\pi}(Q^{2})=0.$ (5)

The second expression is known as von Laue condition [2], which suggests the
force balance inside the pion. It was further conjectured that the
mechanically stable system has a negative $D$-term $D=D(0)<0$ [2]. For the
pion, chiral perturbation theory predicts that in the chiral limit
$D_{\pi}=-1$, the same as free scalar particles [53].

## Gravitational form factor $A_{\pi}$ in light-front holography

The GFF $A_{\pi}(Q^{2})$ can be represented by the LFWFs as [6],

$A_{\pi}(q_{\perp}^{2})=\sum_{n}\int\big{[}{\mathrm{d}}x_{i}{\mathrm{d}}^{2}r_{i\perp}\big{]}_{n}\big{|}\psi_{n}(\\{x_{i},\vec{r}_{i\perp}\\})\big{|}^{2}\sum_{j}x_{j}e^{i\vec{r}_{j\perp}\cdot\vec{q}_{\perp}}$
(6)

where $\psi_{n}(\\{x_{i},\vec{r}_{i\perp}\\})$ is the $n$-body LFWF, and
$\big{[}{\mathrm{d}}x_{i}{\mathrm{d}}^{2}r_{i\perp}\big{]}_{n}$ the $n$-body
phase space measure. Using soft-wall LFWFs of the valence Fock sector
($q\bar{q}$) [54],

$\psi_{\pi}(x,\vec{r}_{\perp})=x(1-x)\varphi_{\pi}(\zeta_{\perp})$ (7)

the valence contribution can be written explicitly as, [55]

$A_{\pi}(Q^{2})=\int{\mathrm{d}}^{2}\zeta_{\perp}\,\big{|}\varphi_{\pi}(\zeta_{\perp})\big{|}^{2}\frac{1}{2}Q^{2}\zeta_{\perp}^{2}K_{2}(\zeta_{\perp}Q)+\cdots$
(8)

The ellipses indicate higher Fock sector contributions, which are important at
low resolution $Q\lesssim\Lambda_{\textsc{qcd}}$. Comparing this expression
with GFF $A_{\pi}$ obtained directly in AdS/QCD [42], one identifies
$(1/2)Q^{2}\zeta_{\perp}^{2}K_{2}(\zeta_{\perp}Q)$ as the UV part of the
dressed current, $H(Q^{2},z)=(1/2)z^{2}Q^{2}K_{2}(zQ)+\cdots$ [55]. Thus, the
high Fock sector contributions can be absorbed into the dressed current,

$A_{\pi}(Q^{2})=\int{\mathrm{d}}^{2}\zeta_{\perp}\,\big{|}\varphi^{2}_{\pi}(\zeta_{\perp})\big{|}^{2}H(Q^{2},\zeta_{\perp})$
(9)

In the soft-wall model, this current is attained from the bulk-to-boundary
propagator of a tensor field [43],

$H(Q^{2},z)=\Gamma\Big{(}2+\frac{a}{2}\Big{)}U\Big{(}\frac{a}{2},-1;2\xi\Big{)}$
(10)

where, $a=Q^{2}/(4\kappa^{2}),\xi=\kappa^{2}z^{2}$, $\Gamma(z)$ is Euler’s
Gamma function, and $U(a,b;z)$ is the second Kummer function [56]. These
results, originally obtained in Ref. [43] for soft-wall AdS/QCD and in Ref.
[55] for HLFQCD, are reproduced in Fig. 1 in comparison with the recent
Lattice prediction [57].

Figure 1: Pion GFFs from HLFQCD as compared with the recent Lattice QCD
results with a pion mass 170 MeV [57]. (Top) $A_{\pi}(Q^{2})$; (Bottom)
$D_{\pi}(Q^{2})$. The soft-wall scale parameter is
$\kappa=0.388\,\mathrm{GeV}$.

## Gravitational form factor $D_{\pi}$ in light-front holography

In contrast to $T^{++}$ (hence GFF $A_{\pi}$), $T^{+-}$ is the conserved
current of the light-front Hamiltonian $P^{-}$, which contains interactions.
To preserve the von Laue condition (5), a consistent treatment of the
interaction is required. Ref. [58] computed the nucleon $D(Q^{2})$ by adopting
the free EMT operator with an effective LFWF that corresponds to the dressed
vector current $J^{\mu}$ in soft-wall AdS/QCD. The obtained GFF $D(Q^{2})$
does not satisfy the von Laue condition (5).

On the AdS/QCD side, GFFs $A(Q^{2})$ can be extracted by coupling the hadronic
fields to a transverse and traceless gravitational field in 5D AdS space [40,
42, 43]. Unfortunately, the same procedure does not access the $D$-term. In a
series of recent works [59, 60, 61], the authors found that the nucleon GFF
$D(Q^{2})$ vanishes in the leading order of $O(1/N_{c})$, and was proportional
to GFF $A(Q^{2})$ in the next to leading order with the proportionality $D(0)$
undetermined. Ref. [62] computed the nucleon $D(Q^{2})$ in the Sakai-Sugimoto
model [63, 64], a top-down model in 10D whose low-energy dual closely
resembles QCD. The authors observed contributions from scalar ($0^{++}$) and
tensor ($2^{++}$) glueballs.

Here, we start from the QCD side with a LFWF representation. This approach is
closely related to the pQCD analysis [65, 66]. We split the EMT operator into
the kinetic energy part and potential energy (interaction) part
$T^{+-}=T^{+-}_{0}+T^{+-}_{\text{int}}$, which corresponds to Hamiltonians,

$P^{-}_{0}=\int{\mathrm{d}}^{3}x\,T^{+-}_{0}(x),\quad
P^{-}_{\text{int}}=\int{\mathrm{d}}^{3}x\,T^{+-}_{\text{int}}(x).$ (11)

The pionic matrix element of $T^{+-}_{0}$ admits an exact LFWF representation
as [7],

$T(Q^{2})\equiv\langle
p+q|T^{+-}_{0}|p\rangle=\int\big{[}{\mathrm{d}}x_{i}{\mathrm{d}}^{2}r_{i\perp}\big{]}_{n}\psi_{n}^{*}(\\{x_{i},\vec{r}_{i\perp}\\})\\\
\times\sum_{j}e^{i\vec{r}_{j\perp}\cdot\vec{q}_{\perp}}\frac{-\nabla_{j\perp}^{2}+m_{j}^{2}-\frac{1}{4}q_{\perp}^{2}}{x_{j}}\psi_{n}(\\{x_{i},\vec{r}_{i\perp}\\})$
(12)

For the interaction part $V(Q^{2})\equiv\langle
p+q|T^{+-}_{\text{int}}|p\rangle$, energy conservation implies

$T(0)+V(0)=2M^{2}_{\pi}=0$ (13)

This expression is equivalent to the von Laue condition (5). To preserve (5)
or (13), it is necessary to adopt an operator $T^{+-}_{\text{int}}$ in
accordance with the hadronic LFWFs. For the soft-wall LFWF, the corresponding
light-front potential energy operator $P^{-}_{\text{int}}$ is attained by
second quantizing the soft-wall potential
$U_{\textsc{sw}}(x,r_{\perp})=\kappa^{2}(\kappa^{2}\zeta^{2}_{\perp}-2)$ where
$\zeta_{\perp}=\sqrt{x(1-x)}r_{\perp}$. $T^{+-}_{\text{int}}(x)$ is its local
density operator, which can be obtained by localizing the light-front
quantized operator $P^{-}_{\text{int}}$ in the transverse direction. The
resulting potential energy density in the valence sector is,

$\widetilde{V}(x_{\perp})=\frac{1}{2}\int\frac{{\mathrm{d}}x}{4\pi
x(1-x)}\int{\mathrm{d}}^{2}r_{\perp}\big{|}\psi(x,r_{\perp})\big{|}^{2}U_{\textsc{sw}}(x,r_{\perp})\\\
\times\Big{\\{}\delta^{2}\big{(}x_{\perp}-(1-x)r_{\perp}\big{)}+\delta^{2}(x_{\perp}+xr_{\perp})\Big{\\}}$
(14)

Plugging in the soft-wall wave function (7), the pion GFF $D_{\pi}(Q^{2})$
reads,

$D_{\pi}(Q^{2})=\int{\mathrm{d}}^{2}\zeta_{\perp}\big{|}\varphi_{\pi}(\zeta_{\perp})\big{|}^{2}\Big{\\{}\frac{\zeta_{\perp}^{2}Q^{2}}{4}K_{2}(\zeta_{\perp}Q)-2K_{0}(\zeta_{\perp}Q)\\\
-\frac{2U_{\textsc{sw}}(\zeta_{\perp})}{Q^{2}}\Big{[}\zeta_{\perp}QK_{1}(\zeta_{\perp}Q)-\frac{1}{2}\zeta^{2}_{\perp}Q^{2}K_{2}(\zeta_{\perp}Q)\Big{]}\Big{\\}}$
(15)

where $K_{\nu}(z)$ is the modified Bessel function of the second kind [56],
$U_{\textsc{sw}}(\zeta_{\perp})=\kappa^{2}(\kappa^{2}\zeta_{\perp}-2)$ is the
(soft-wall) holographic potential. It is not hard to see this expression
satisfies the von Laue condition (5) as expected. Eq. (15) is the main result
in this note. Fig. 1 shows Eq. 15 (bare current) in comparison with the recent
Lattice calculation [57]. At large $Q^{2}$, the form factor scales as
$1/Q^{2}$, in agreement with the pQCD analysis [66].

Similar to Eq. (8), the valence ansatz captures physics at large $Q^{2}$. In
the forward limit $Q^{2}\to 0$, the obtained $D_{\pi}(Q^{2})$ diverges. To
better incorporate physics beyond the valence Fock sector, we can dress the
current. One readily identifies $\zeta_{\perp}QK_{1}(\zeta_{\perp}Q)$ and
$\frac{1}{2}\zeta_{\perp}^{2}Q^{2}K_{2}(\zeta_{\perp}Q)$ as the UV parts of
the dressed vector and tensor currents [30]

$\displaystyle V(Q^{2},\zeta_{\perp})=\Gamma(1+a)U(a,0;\xi),$ (16)
$\displaystyle
H(Q^{2},\zeta_{\perp})=\Gamma(2+\frac{a}{2})U(\frac{a}{2},-1;2\xi),$ (17)

respectively with $a=Q^{2}/(4\kappa^{2})$ and
$\xi=\kappa^{2}\zeta_{\perp}^{2}$. $K_{0}(\zeta_{\perp}Q)$ may be identified
as the bare scalar current. In principle, the dressing of this current can be
obtained from the bulk-to-boundary propagator of the scalar fields. The IR
finite bulk-to-boundary propagators of the scalar field with conformal
dimension $\Delta=3$ (i.e. scalar mesons) and $\Delta=4$ (i.e. scalar
glueballs) in soft-wall model are
$S_{\Delta=3}=z\Gamma(a+\frac{3}{2})U(a+\frac{1}{2},0,\xi)$ and
$S_{\Delta=4}=\Gamma(a+2)U(a,-1,\xi)$, respectively [67, 68, 69]. We assume
both currents contribute to the $D$-term,

$S(Q^{2},\zeta_{\perp})=c_{1}S_{\Delta=3}(Q^{2},\zeta_{\perp})+c_{2}S_{\Delta=4}(Q^{2},\zeta_{\perp})$
(18)

and their coefficients $c_{1}=0.79,c_{2}=0.25$ are determined from matching to
both the UV asymptotics (15) and the IR limit $D_{\pi}(0)=-1$ as predicted by
the chiral perturbation theory [53]. The resulting GFF $D_{\pi}(Q^{2})$ with
the dressed currents is,

$D_{\pi}(Q^{2})=\int{\mathrm{d}}^{2}\zeta_{\perp}\big{|}\varphi_{\pi}(\zeta_{\perp})\big{|}^{2}\Big{\\{}\frac{1}{2}H(Q^{2},\zeta_{\perp})-2S(Q^{2},\zeta_{\perp})\\\
-\frac{2U_{\textsc{sw}}(\zeta_{\perp})}{Q^{2}}\Big{[}V(Q^{2},\zeta_{\perp})-H(Q^{2},\zeta_{\perp})\Big{]}\Big{\\}}.$
(19)

Fig. 1 compares results with the dressed current (19) and the bare current
(15), together with the recent Lattice calculation [57] (cf. Ref. [70]). The
dressed current are in good agreement with with recent Lattice simulation at
$Q^{2}\lesssim 1\,\mathrm{GeV}^{2}$. From these results, we can extract the
mechanical radius of the pion,
$r_{\text{mech}}^{2}\equiv-6D^{\prime}(0)/D(0)=(0.60\,\mathrm{fm})^{2}$, which
is larger than the predicted pion matter (mass) radius
$r^{2}_{\text{mat}}\equiv-6A^{\prime}(0)/A(0)=(0.39\,\mathrm{fm})^{2}$. These
values are very close to the corresponding radii $r^{2}_{\text{mat}}=(0.41\pm
0.01\,\mathrm{fm})^{2}$ and $r_{\text{mech}}^{2}=(0.61\pm
0.07\,\mathrm{fm})^{2}$ extracted from the recent Lattice simulation [57],
while differ from the mechanical radius
$r_{\text{mech}}^{2}=(0.82-0.88\,\mathrm{fm})^{2}$ extracted from the two
photon process $\gamma^{*}\gamma\to\pi^{0}\pi^{0}$ by $\sim$30% [71].

## Summary and discussions

In this note, we investigated the pion gravitational form factors by
exploiting the remarkable correspondence between light-front QCD and gravity
theory in 5D. We computed the $D$-term of the pion using the bare EMT on the
light front valid at high resolution. To extend the results to low resolution
probes, we adopted scalar, vector and tensor dressed currents based on bulk-
to-boundary propagators for scalar, vector and tensor fields.

The $D$-term of the pion stems from several sources. The largest contribution
stems from the coupling to the scalar mesons, $D_{\pi}^{M}(0)=-0.83$. This
current contains a series of scalar meson poles $M_{n}^{2}=(4n+6)\kappa^{2}$
in the timelike region [69],

$D_{\pi}^{M}(-q^{2})=\sum_{n=0}\frac{f_{n}}{q^{2}-M_{n}^{2}}.$ (20)

The lowest scalar meson mass $M=\sqrt{6}\kappa=0.95\,\mathrm{GeV}$ is close to
the mass of $a_{0}(980)$. The scalar glueball contributes to a pole term

$D_{\pi}^{G}(-q^{2})=\frac{4\kappa^{2}}{q^{2}-8\kappa^{2}}$ (21)

The pole mass $M_{G}=\sqrt{8}\kappa=1.1\,\mathrm{GeV}$ is the mass of the
ground-state scalar glueball predicted in AdS/QCD. The corresponding $D$-term
is $D_{\pi}^{G}(0)=-1/2$. The contribution of the tensor current has an
opposite sign, $D_{\pi}^{T}(0)=+1/2$.

The contribution of the last term in (19) is small, $D_{\pi}^{V-T}(0)=-0.17$.
Chiral symmetry plays a pivotal role here. In the hard-wall model, the
corresponding term vanishes completely. For other hadrons, the emergent hadron
mass is expected to give a large contribution that significantly alters the
budget of the $D$-term. This term consists of a vector current and a tensor
current. However, in contrast to the nucleon case, the $\rho$-pole at
$M^{2}_{\rho}=4\kappa^{2}$ does not contribute to the $D$-term. In light-front
dynamics, according to pQCD dimensional counting rules [72, 73], the vector
current contains a twist-3 ($|q\bar{q}g\rangle$) contribution and a twist-2
contribution ($|q\bar{q}\rangle$). It is interesting to see the net result is
the twist-3 contribution minus the twist-2 result, leaving the pure gluon
contribution. This cancellation relies critically on the fact that the pion is
massless in the chiral limit. Its remarkable that the particle identification
in AdS/QCD is consistent with the pQCD counting rule [72, 73].

We note that the picture presented here is qualitatively similar to the recent
holographic QCD computation of the nucleon GFF $D(Q^{2})$ within the Sakai-
Sugimoto model in 10 dimensions [62]. Furthermore, the method we proposed here
can also be extended to the nucleon sector and be compared with the results
from these models directly.

## Acknowledgements

The authors acknowledge fruitful discussions with Stan Brodsky and Guy de
Téramond. Y.L. is supported by the New faculty start-up fund of the University
of Science and Technology of China. This work was supported in part by the
National Natural Science Foundation of China (NSFC) under Grant No. 12375081,
by the Chinese Academy of Sciences under Grant No. YSBR-101, and by the US
Department of Energy (DOE) under Grant No. DE-SC0023692.

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