---
abstract: 'We consider an elastic rod model for twisted DNA in the plectonemic regime. The molecule is treated as an impenetrable tube with an effective, adjustable radius. The model is solved analytically and we derive formulas for the contact pressure, twisting moment and geometrical parameters of the supercoiled region. We apply our model to magnetic tweezer experiments of a DNA molecule subjected to a tensile force and a torque, and extract mechanical and geometrical quantities from the linear part of the experimental response curve. These reconstructed values are derived in a self-contained manner, and are found to be consistent with those available in the literature.'
author:
- 'N. Clauvelin'
- 'B. Audoly'
- 'S. Neukirch'
bibliography:
- 'dna\_pre.bib'
title: |
    Mechanical response of plectonemic DNA:\
    an analytical solution
---

Introduction
============

Mechanical properties of the DNA molecule play an important role in the biological processes involved in the cell, yet we only have an imprecise view of these properties. Advances in nanotechnologies make it possible to exert forces onto isolated DNA filaments: mechanical response of the molecule is now widely studied. Single molecule experiments provide a powerful way to investigate the behavior of DNA subjected to mechanical stress. In such experiments, the molecule is held by optical or magnetic tweezers and forces and torques are applied to it [@smith+al:1992; @bustamente+al:1994]. The interaction between DNA and proteins is actively investigated; for instance, the chemical and mechanical action of an enzyme on a molecule can be inferred from the global deformation of the molecule [@RevyakinLiuEbright-Abortive-Initiation-and-Productive-Initiation-by-RNA-Polymerase-2006].

In this paper we focus on a specific type of experiments: a double stranded DNA molecule is fixed by one end to a glass surface while the other end is attached to a magnetic bead; using a magnet, a pulling force and a torque are applied on the DNA filament [@strick+al:1996]. Large ranges of pulling forces, from one tenth to one hundred piconewton, and number of turns can be explored in the experiments, and the molecule displays a variety of behaviors and conformations [@allemand+al:1998; @sarkar+al:2001; @bryant+al:2003]. We study the response of the molecule to moderate forces, below 10 pN, and moderate to large number of turns, equivalent to a positive supercoiling ratio of the order of 0.1. In experiments, the pulling force is kept constant while the bead is rotated gradually. Above a threshold value of the number of turns, the molecule wraps around itself in a helical way, giving rise to a structure comprising plectonemes. The vertical extension of the molecule is recorded and plotted as a function of the number of turns. Experimental rotation-extension curves have a characteristic shape and are called *hat curves* [@bouchiat_2000; @strick_1998]. The central, rounded part of the curve can be explained using the worm-like chain (WLC) model [@marko+siggia:1995a] and its variants. At larger number of turns, the extension of the molecule decreases linearly. This linear part is obtained when the molecule is in supercoiled configuration and forms plectonemes. The plectonemic structure is made of two interwound helical filaments whose geometry is characterized by the so-called superhelical angle and radius; note that each of these filaments is itself made of a double-stranded DNA molecule. The superhelical angle and the twisting moment in the filaments are key parameters that control the action of topoisomerases [@koster+al:2005], RNA polymerase [@revyakin+al:2004], or other enzymes [@heijden+al:2005] on DNA. The distance of self-approach of DNA in supercoiled regime has been the subject of a number of studies [@bednar+al:1994; @rybenkov_1997; @rybenkov+al:1993; @charvin+al:2005]. In previous analytical and numerical work, the double stranded DNA molecule has been modelled as a twist-storing elastic filament. These approaches have been successful at reproducing the response of DNA to moderate torque [@bouchiat_1998; @moroz+nelson:1997], given by the central region of the experimental curves. The analysis of the linear regions of these curves, based on a detailed model of plectonemes, was lacking until recently: in Ref. [@marko:2007], a composite model based on an empirical free energy of supercoiled DNA is proposed.

Here we present an elastic rod model for helical supercoiling of the DNA molecule, which is relevant to a large number of turns. Our model is self-contained and provides an mechanically accurate description of elastic filaments in contact. The molecule is divided in two domains: one where the configuration is a worm-like-chain, dominated by thermal fluctuations, and the other one, a superhelical region dominated by elasticity, where the molecule contacts itself. The plectonemic regions can be spread in various places of the molecule; as this does not change the mechanical response of the system, we refer to these regions as if they were in one chunk. We deal with self-contact by introducing an effective superhelical radius (distinct from the crystallographic radius of 1 $\textrm{nm}$, from the size of the Manning condensate and from the Debye length, although in the same range of values), which varies with external loads and salinity of the solution. The effective radius is defined as the radius of a chargeless, impenetrable and elastic tube having the same mechanical response as the molecule. This radius is not given as a parameter of the model and is extracted from experimental data. Using an energy approach, we relate geometrical variables (superhelical radius and angle) to applied force and torque. We also characterize the response of the molecule in the plectonemic regime, extend former numerical results [@neukirch_2004], and show how geometrical and mechanical parameters can be extracted from experimental data.

Model
=====

The present model investigates the equilibrium behavior of an elastic rod with bending rigidity $K_0$ (the bending persistence length is $A=K_0/(kT)$, where $k$ is the Boltzmann constant and $T$ the absolute temperature) and twisting rigidity $K_3$ under traction and torsion as shown in Fig. \[fig:ExperimentSetup\]. This is a coarse-grained model for DNA where base-pairs details are neglected. For instance, the anisotropic flexibility of the molecule, originating from base pairing and major-minor grove geometry, is smoothed out at a scale of several base pairs: a highly twisted anisotropic rod can be replaced by an equivalent isotropic rod with effective bending rigidity [@kehrbaum_2000].

Geometry {#geometry .unnumbered}
--------

We start with a geometric description of the rod configurations relevant to the plectonemic regime. This defines a reduced set of configurations (Ansatz), over which we shall minimize the elastic strain energy associated with deformations. The rod, of length $\ell$, is considered inextensible and has circular cross-section; let $s$ denote the arclength along the rod. The strain energy involves, at lowest order, the geometric curvature $\kappa(s)$ of the centerline of the rod as well as the twist $\tau(s)$. We emphasize that the twist $\tau(s)$ is different from the geometrical (Frénet-Serret) torsion of the centerline as it takes into account the rotation of material cross sections around the centerline. It allows one to distinguish between twisted and untwisted configurations of the rod having the same centerline. The rod centerline is parameterized by $\mathbf{r}(s)$ and its unit tangent $\mathbf{t} {\stackrel{\mbox{\rm\scriptsize def}}{=}}{\mathrm{d}}\mathbf{r} / {\mathrm{d}}s $ can be described with spherical angles, as shown in Fig. \[fig:ExperimentSetup\]: $\alpha(s)$ is the zenith angle and $\psi(s)$ the azimuth angle with respect to the direction $\mathbf{e_x}$ along the common axis of the two superhelices in the plectonemic region.

![Sketch of the magnetic tweezers experiment. A B-DNA molecule of total contour length $\ell$ is fixed in $s=0$ to a glass surface while the other end in $s=\ell$ is attached to a magnetic bead. A pulling force $F_{\mathrm{ext}}$ and a torque $M_{\mathrm{ext}}$ are applied at the upper end by using a magnet. The superhelical angle and radius are denoted $\alpha$ and $R$ respectively.[]{data-label="fig:ExperimentSetup"}](SchemaPlecto.eps){width="9.cm"}

We consider the following configurations, relevant to a large applied number of turns, $n$. The tails are assumed to be straight but twisted (thermal fluctuations will be accounted for by using the rescaled tail length predicted by WLC theory). The plectonemes are described by two identical and uniform helices where, again, each of these helices is itself a double-stranded DNA molecule. Both the end loop of the plectonemes and the matching region between the tails and the plectonemic part are neglected.

In the tails the rod is straight and aligned with the $\mathbf{e_z}$ axis, and $\mathbf{t}=\mathbf{e_{z}}$ there. The geometric curvature $\kappa {\stackrel{\mbox{\rm\scriptsize def}}{=}}\left| {\mathrm{d}}\mathbf{t} / {\mathrm{d}}s \right|$ is zero, $\kappa(s)=0$.

In each filament of the plectonemes, the position vector $\mathbf{r}(s)$ and the tangent vector $\mathbf{t}(s)$ describe a superhelix of axis $\mathbf{e_x}$: $$\begin{cases}
        r_x(s)=s\,\cos\alpha \\
        r_y(s)=\chi \,R\,\sin\psi(s) \\
        r_z(s)=-\chi \,R\,\cos\psi(s)
    \end{cases}
    \; \text{and} \;\;
    \begin{cases}
        t_x(s)=\cos\alpha \\
        t_y(s)=\sin\alpha\,\cos\psi(s) \\
        t_z(s)=\sin\alpha\,\sin\psi(s) 
    \end{cases}
    \label{eqn:PosVector-Plecto}$$ The other filament of the plectonemes is obtained by a rotation of $180^\circ$ around the axis $\mathbf{e_{x}}$. Here $\chi=\pm 1$ stands for the chirality of the two helices and $R$ and $\alpha$ denote the superhelical radius and angle, respectively. In equation (\[eqn:PosVector-Plecto\]), the condition $\mathrm{d}\mathbf{r}/\mathrm{d}s = \mathbf{t}$ yields $ {\mathrm{d}}\psi / {\mathrm{d}}s=\chi \, \sin\alpha / R $. The curvature in the plectonemes is $\kappa(s) {\stackrel{\mbox{\rm\scriptsize def}}{=}}\left| {\mathrm{d}}\mathbf{t} / {\mathrm{d}}s \right| = \frac{\sin^2\alpha}{R}$.

Noting $\ell_{\mathrm{p}}$ the contour length spent in the plectonemes, we obtain the following expression for the integral of the squared curvature over the whole length of the rod: $$\int_{0}^{\ell} \kappa^2\,\mathrm{d}s = 
    \frac{\sin^4\alpha}{R^2}\,
    \ell_{\mathrm{p}}
    \textrm{.}
    \label{eq:IntegralOfSquareCurvature}$$

The end torque twists the filament. For a rod with circular cross-section, the twist $\tau(s)$ at equilibrium is uniform [@antman_2005], ${\mathrm{d}}\tau/ {\mathrm{d}}s = 0$ for all $s$. As a result, the equilibrium configuration of the rod is fully specified by the centerline, through the variables $\alpha$, $R$ and $\ell_{\mathrm{p}}$, and an additional scalar $\tau$ describing twist.

The twist parameter $\tau$ is geometrically related to the number of turns imposed on the magnetic bead, $n$, which is equal to the link of the DNA molecule, $n=\mathrm{Lk}$. In the present case the link reads [@neukirch_2004]: $$\mathrm{Lk} = \mathrm{Tw}+\mathrm{Wr} =
    \frac{1}{2\pi}\int^\ell_0\tau\,{\mathrm{d}}s -
    \chi\,\frac{\sin2\alpha}{4\pi\,R}\,\ell_p
    =\frac{1}{2\pi}\,\left(
    \tau\,\ell
    -\chi\,\frac{\sin2\alpha}{2\,R}\,\ell_{\mathrm{p}}
    \right),
    \label{eqn:LinkDefinition}$$ as we neglect the writhe in the tails.

Energy formulation {#energy-formulation .unnumbered}
------------------

Using the above notations the rod is described by four variables: $\alpha$ the superhelical angle, $R$ the superhelical radius, $\tau$ the twist and $\ell_p$ the contour length spent in the plectonemes. We proceed to derive the total energy of the system as a function of these four variables. It is the sum of three terms, $V=V_{\mathrm{el}}+V_{\mathrm{ext}} + V_{\mathrm{int}}$, where the first term is the strain elastic energy, the second is the potential energy associated with the external loads $F_{\mathrm{ext}}$ and $M_{\mathrm{ext}}$, and the third accounts for interaction of the filaments in the plectonemes. The strain elastic energy for the rod of total contour length $\ell$ is : $$V_{\mathrm{el}}=\frac{K_0}{2}\int^{\ell}_{0}\kappa^2\,{\mathrm{d}}s+\frac{K_3}{2}\int^{\ell}_{0}\tau^2\,{\mathrm{d}}s~.
    \label{eqn:ElasticEnergy}$$ We do not take into account the reduction of the effective torsional rigidity in the tails due to fluctuations [@moroz+nelson:1997]. The potential energy is given by: $$V_{\mathrm{ext}}=-F_{\mathrm{ext}} (z(\ell)-z(0)) -
    2\pi\,M_{\mathrm{ext}}\,n~,
    \label{eqn:LoadsEnergy}$$ where $z(\ell)-z(0)=\ell - \ell_p$ for straight tails and $n=\mathrm{Lk}$.

If the DNA-DNA interaction was clearly established, we would include the corresponding interaction energy $V_{\mathrm{int}}$ in the total energy $V$ [@CAN_U_electro]. This is not the case and we model the filaments in electrostatic interaction as effective chargeless hard-core tubes. The effective radius $a$ of these tubes accounts for a variety of physical mechanisms and in particular for the presence of counter-ions. As in Refs. [@VologodskiiCozzarelli-Modeling-of-long-range-electrostatic-interactions-in-DNA-1995; @rybenkov_1997], we do not fix the quantity $a$ in advance and let it vary with experimental conditions, such as applied load and salinity. In fact, we show how $a$ can be extracted from experimental measurements. Doing so, we replace the actual (unknown) interaction potential $V_{\mathrm{int}}(R,\alpha)$ by a hard-core interaction with adjustable radius $a$, and optimize $a$ to best fit a given experiment.

The parameter $a$ must certainly be larger than the crystallographic DNA radius $1~\mathrm{nm}$. It is different from the radius of the Manning condensate [@manning_1969_1; @manning_1969_2; @manning_1969_3] since approximately a quarter of the charge remains outside of the Manning condensate. The equilibrium is the solution of a constrained minimization problem for the elastic energy, subjected to the impenetrability condition $$R\geq a
    \textrm{.}
    \label{eq:RLargerThanA}$$ We anticipate on the fact that there is contact, $R=a$, for typical experimental conditions. This constraint is handled by a Lagrange multiplier $\lambda$; the actual interaction energy is then substituted with the following expression: $$V_{\mathrm{int}}=-\lambda\left(R-a\right)
    \textrm{.}
    \label{eqn:InteractionEnergy}$$

Combining Eqs. (\[eq:IntegralOfSquareCurvature\]–\[eqn:InteractionEnergy\]), we write the total potential energy of the system as: $$\begin{gathered}
    V(\alpha, R, \tau, \ell_{\mathrm{p}}) =
    \frac{K_0}{2}\,\frac{\sin^4\alpha}{R^2}\,\ell_p +
    \frac{K_3}{2}\,\tau^2\,\ell
    - F_{\mathrm{ext}}\left(\ell-\ell_\mathrm{p}\right) 
    \\ {}-M_{\mathrm{ext}}\,\left(
    \tau\,\ell
    -\chi\,\frac{\sin2\alpha}{2\,R}\,\ell_{\mathrm{p}}
    \right)
    - \lambda\left(R-a\right)
    \textrm{.}
    \label{eqn:TotalEnergy}
    \end{gathered}$$ In Ref. [@kutter_2001] a similar energy function has been introduced but the rest of analysis differs from ours. Indeed, their approach focuses on statistical mechanics and the analysis of the state of lowest energy is overlooked. Moreover, the parameter $a$ is fixed a priori to the crystallographic radius of DNA, $a=1~\mathrm{nm}$, which is a strong underestimation of the actual distance of self-approach of DNA in saline solution. In contrast, we undertake a detailed analysis of the equilibrium solutions, with thermal fluctuations considered in the tails; this allows us to derive simple formulas for the force and the moment as a function of the superhelical variables, applicable to magnetic tweezers experiments.

Results
=======

Mechanical equilibrium is given by the Euler-Lagrange condition for the stationarity of the potential $V(\alpha,
R,\tau,\ell_{\mathrm{p}})$ in Eq. (\[eqn:TotalEnergy\]) with respect to its variables, $$\left( 
    \frac{\partial V}{\partial \tau},\, 
    \frac{\partial V}{\partial \alpha},\,
    \frac{\partial V}{\partial \ell_\mathrm{p}},\, 
    \frac{\partial V}{\partial R} \right)=
    0.$$

The first condition $\partial V/\partial \tau$ allows one to recover the constitutive relation for twist deformations, $M_{\mathrm{ext}} =
K_{3}\, \tau$, given that the twisting moment is uniform in the filament and equal to the applied torque $M_{\mathrm{ext}}$.

Variation of the total energy with respect to $\alpha$ gives the expression of the applied torque $M_{\mathrm{ext}}$ in terms of the superhelical variables $\alpha$ and $R$: $$M_{\mathrm{ext}} =
    -\frac{2\chi\,K_0}{R}\frac{\cos\alpha\,\sin^3\alpha}{\cos2\alpha}
    \textrm{,}
    \label{eqn:VarTorsionalMoment}$$ which is what was found for purely plectonemic solution (no tails) [@thompson+al:2002].

The condition $\partial V/\partial \ell_{\mathrm{p}} = 0$, combined with Eq. (\[eqn:VarTorsionalMoment\]), allows one to relate the pulling force $F_{\mathrm{ext}}$ to the superhelical geometry: $$F_{\mathrm{ext}} =
    \frac{K_0}{R^2}\,\sin^4\alpha\,\left(\frac{1}{2}+\frac{1}{\cos 2\alpha}\right)
    \textrm{.}
    \label{eqn:VarPullingForce}$$ This formula justifies and extends the numerical fit $F_{\mathrm{ext}}
\propto K_{0}\,\alpha^4/R^2$ found in Ref. [@neukirch_2004] for small values of $\alpha$.

The Euler-Lagrange condition with respect to $R$ yields an equation involving the Lagrange multiplier $\lambda$. The quantity $\lambda /
\ell_{\mathrm{p}}$ can be interpreted as the contact force per unit length, $p$, of one filament onto the other. Eqs. (\[eqn:TotalEnergy\]–\[eqn:VarTorsionalMoment\]), together with the condition $\partial V/\partial R=0$, yields: $$p=\frac{\lambda}{\ell_p}=\frac{K_0}{R^3}\frac{\sin^4\alpha}{\cos2\alpha}
\textrm{.}
    \label{eqn:VarContactPressure}$$ Note that this pressure (more accurately, force per unit length) is positive for $\alpha\leq \pi/4$; if our assumption of contact $R=a$ was incorrect, this would be indicated by a negative pressure value here.

In magnetic tweezers experiments, the pulling force $F_{\mathrm{ext}}$ is imposed although the applied torque $M_{\mathrm{ext}}$ is unknown. The two unknowns $R$ and $\alpha$ are then related by Eq. (\[eqn:VarPullingForce\]); in the next Section, a second equation relating those unknowns and the extension $z$ is given, which makes it possible to solve for $R$ and $\alpha$. The twisting moment can then be found from Eq. (\[eqn:VarTorsionalMoment\]).

Vertical extension of the filament {#vertical-extension-of-the-filament .unnumbered}
----------------------------------

In magnetic tweezers experiments, the measurable quantities are the vertical extension $z$ and the number of turns $n$ imposed on the bead. Using Eq. (\[eqn:LinkDefinition\]) for $n=\mathrm{Lk}$, the equation $z = \ell-\ell_\mathrm{p}$ and the constitutive relation $\tau =
M_{\mathrm{ext}}/K_{3}$ where $M_{\mathrm{ext}}$ is found from Eq. (\[eqn:VarTorsionalMoment\]), we obtain the vertical extension of the filament as a linear function of the number of turns $n$: $$z =
    \left(1+\frac{2\,K_0}{K_3}\frac{\sin^2\alpha}{\cos2\alpha}\right)\ell
    +\chi\,n\,\frac{4\pi\,R}{\sin2\alpha}~.
    \label{eqn:VerticalExtension}$$ Thermal fluctuations dominantly affect the tails and make the end-to-end distance $z$ of the molecule smaller than the contour length $\ell-\ell_{\mathrm{p}}$ of the tail parts, by a factor $\rho_{\mathrm{wlc}}\in [0,1]$: $z = \rho_{\mathrm{wlc}}\,(\ell -
\ell_\mathrm{p})$. This factor depends on both the pulling force $F_{\mathrm{ext}}$ and the bending persistence length $A=K_0/(kT)$ and can either be read off an experimental hat curve from the value $z(n=0) =
\rho_{\mathrm{wlc}}\,\ell$, or computed from theoretical formulas [@marko+siggia:1995a; @bouchiat+al:1999]. To account for these thermal effects, we replace Eq. (\[eqn:VerticalExtension\]) with: $$z =
\rho_{\mathrm{wlc}}\left(1+\frac{2\,K_0}{K_3}\frac{\sin^2\alpha}{\cos2\alpha}\right)\ell
    +\chi \, \rho_{\mathrm{wlc}}\, \frac{4\pi\,R}{\sin2\alpha} \,n~.
    \label{eqn:VerticalExtensionWLC}$$

One of the main features of the experimental hat curves is the linear decrease of the vertical extension with the number of turns. We define the slope $q$ in the linear part of the hat curve as: $$q {\stackrel{\mbox{\rm\scriptsize def}}{=}}\left| \frac{{\mathrm{d}}z}{{\mathrm{d}}n} \right|=\rho_{\mathrm{wlc}}
    \frac{4\pi\,R}{\sin2\alpha}~.
    \label{eqn:slope hat curve}$$ Given experimental values of $F_{\mathrm{ext}}$ and $q$, Eqs. (\[eqn:VarPullingForce\]) and (\[eqn:slope hat curve\]) can be solved for $R$ and $\alpha$. Since $q$ (and $F_{\mathrm{ext}}$) are constant along the linear part of a hat curve, the values of $R$ and $\alpha$ thus determined will be constant as well. As a result, the twisting moment in the molecule, given by Eq. (\[eqn:VarTorsionalMoment\]), is constant, for a given experiment, along the linear region of the hat curve, a property that has been previously reported in the literature [@bouchiat_2000; @marko:2007] and which is a clear outcome of the present model. An interpretation of the fact that $R$ and $\alpha$ are constant in the linear region of the hat curve is that each additional turn of the bead is used to convert a small piece of tail into plectonemes.

Twisting moment {#twisting-moment .unnumbered}
---------------

The twisting moment in the molecule, which is uniform and equal to $M_{\mathrm{ext}}$ at equilibrium, cannot be measured in magnetic tweezers experiments. However, it has been shown that enzyme activity such as RNA polymerase depends on the value of the twisting moment in DNA [@revyakin+al:2004]. The value of $M_{\mathrm{ext}}$ can be determined from Eq. (\[eqn:VarTorsionalMoment\]) once $R$ and $\alpha$ are known, as explained above. Here, we give a formula for $M_{\mathrm{ext}}$ directly as function of the experimental slope $q$ and the external force $F_{\mathrm{ext}}$. Indeed, using Eq. (\[eqn:slope hat curve\]) to eliminate $R$ in Eqs. (\[eqn:VarTorsionalMoment\]) and (\[eqn:VarPullingForce\]), one obtains $M_{\mathrm{ext}}(q,\alpha)$ and $F_{\mathrm{ext}}(q,\alpha)$ as functions of $q$ and $\alpha$. It is then possible to eliminate $\alpha$, which yields: $$M_{\mathrm{ext}} =
    m+\left( m^2 + 2 K_0 \,F_{\mathrm{ext}} \right)^{1/2}
    ,
    \qquad
    \textrm{where }m=
    \frac{q \,F_{\mathrm{ext}}}{4 \pi\, \rho_{\mathrm{wlc}}} 
    - \frac{3\pi\, \rho_{\mathrm{wlc}} \,K_0}{2q}
    \label{eqn:ForceMoment}$$ In the limit of small $\alpha$, one can expand the functions $M_{\mathrm{ext}}(q,\alpha)$ and $F_{\mathrm{ext}}(q,\alpha)$ prior to elimination of $\alpha$, and this leads to a simplified formula: $$M_{\mathrm{ext}} \simeq \frac{2\,q}{3\,\pi \, \rho_{\mathrm{wlc}}}F_{\mathrm{ext}}
    \textrm{,}
    \label{eqn:ApproxForceMoment}$$ where, as explained above, $\rho_{\mathrm{wlc}} = z(n=0)/\ell$. As shown in Fig. \[fig:TorsionalMoment\], this approximation is accurate when used with typical experimental values. Eq. (\[eqn:ApproxForceMoment\]) provides a simple and direct mean of evaluating the twisting moment in magnetic tweezers experiments, based on the slope of the linear part of the hat curve only. Note that it should not be inferred from Eq. (\[eqn:ApproxForceMoment\]) that $M_{\mathrm{ext}}$ depends linearly on $F_{\mathrm{ext}}$, as the slope $q$ is itself a function of $F_{\mathrm{ext}}$.

Superhelical angle limit {#superhelical-angle-limit .unnumbered}
------------------------

It is known that the topology of contact between two impenetrable helical tubes winding along a common axis changes when $\alpha$ becomes larger than $\pi/4$ [@neukirch+heijden:2002]. The possibility of such a change of topology is not considered in our model (being specific to hard-core repulsion between tubes, it is not relevant to DNA molecules undergoing long-range electrostatic repulsion anyway). Nevertheless, the equilibrium solutions found here are all such that $\alpha < \pi/4$. This upper bound has a mechanical origin, and not a geometrical one: the expressions for $F_{\mathrm{ext}}$ in Eq. (\[eqn:VarPullingForce\]) and for $M_{\mathrm{ext}}$ in Eq. (\[eqn:VarTorsionalMoment\]) both diverge at $\alpha=\pi/4$ and plectonemic solutions with a superhelical angle larger than $\pi/4$ are unstable.

Application to experiments {#application-to-experiments .unnumbered}
--------------------------

The model is used to extract mechanical and geometrical parameters from experimental data. To allow comparison with previous work, we use the same data as in [@neukirch_2004]. These data are shown in Fig. \[fig:HatCurves\]; they were obtained on a 48kbp lambda phage DNA molecule in a 10mM phosphate buffer.

![Experimental hat curves showing the vertical extension of a lambda phage DNA 48kbp molecule as a function of the number of turns imposed on the magnetic bead (salt concentration $10\,\mathrm{mM}$, temperature $298\,K$). Experimentally measured persistence length of the molecule is $A=51.35\,\mathrm{nm}$. Each curve corresponds to a fixed pulling force $F_{\mathrm{ext}}$: $0.25$, $0.33$, $0.44$, $0.57$, $0.74$, $1.10$, $1.31$, $2.20$, $2.95\,\mathrm{pN}$. Triangles represent the fit for the slope $q$ of the linear region. Data kindly provided by V. Croquette (LPS – ENS, Paris).[]{data-label="fig:HatCurves"}](HatCurves.eps){width="9.cm"}

For each curve in Fig. \[fig:HatCurves\], corresponding to a given value of the external force $F_{\mathrm{ext}}$, we extract the slope $q$ by fitting the linear region. The superhelical variables $R$ and $\alpha$ are found by solving Eqs. (\[eqn:VarPullingForce\]) and (\[eqn:slope hat curve\]), and are plotted in Fig. \[fig:PlectonemicVariables\] as a function of $F_{\mathrm{ext}}$. The reconstructed values of $R$ are in the nanometric range; they decrease with the pulling force, from approximately 6 to 2 times the DNA crystallographic radius in this particular experiment. At large values of the force, $R$ is close to (and actually smaller than) the Debye length, $3.07\;\mathrm{nm}$ in $10\;\mathrm{mM}$ salt, and the Manning condensation radius, $3.18\;\mathrm{nm}$ in $10\;\mathrm{mM}$ salt [@shaughnessy_2005]. We note that the values of $R$ found here in the presence of a pulling force are smaller than (and in the same range as) in Ref. [@rybenkov+al:1993] where no force is applied, which is consistent. The reconstructed values of the twisting moment $M_{\mathrm{ext}}$ and of the contact pressure $p$ are given in Fig. \[fig:TorsionalMoment\], based on the same experimental data. The values of $M_{\mathrm{ext}}$ are determined both by Eq. (\[eqn:VarTorsionalMoment\]) using the previously computed values of $R$ and $\alpha$, and by the approximate formula (\[eqn:ApproxForceMoment\]) directly. A good agreement is obtained, which validates the proposed approximation. The values of $M_{\mathrm{ext}}$ are also compared to those predicted by a composite analytical model, see Eq. (17) in Ref [@marko:2007] (this model uses effective parameters determined from Monte-Carlo simulations [@vologodskii+marko:1997]).

![Reconstructed values of the plectonemic radius $R$ as a function of the pulling force, from the data in Fig. \[fig:HatCurves\] by solving Eqs. (\[eqn:VarPullingForce\]) and (\[eqn:slope hat curve\]). The angle $\alpha$ is shown in the inset.[]{data-label="fig:PlectonemicVariables"}](Radius+Angle.eps){width="9.cm"}

![Reconstructed values for the twisting moment in the molecule based on the data shown in Fig. \[fig:HatCurves\], using the exact formula in Eq. (\[eqn:VarTorsionalMoment\]) (solid squares), and the small angles approximation in Eq. (\[eqn:ApproxForceMoment\]) (open circles). Comparison with the prediction of the composite model in Ref. [@marko:2007] (curve). Contact pressure is shown in inset.[]{data-label="fig:TorsionalMoment"}](Moment+Pression.eps){width="9.cm"}

Conclusion
==========

We have shown that, under the approximation that thermal fluctuations are neglected in the plectonemes, one can calculate analytically the response of twisted DNA: supercoils are described by a mechanically exact and self-contained model. Self-contact in the plectonemic region is treated with a hard-core potential; an expression for the contact pressure between the two dsDNA is derived. The hard-core radius is an effective parameter determined, for a given value of the applied force, from the slope of the linear region of the experimental curve. A formula for the twisting moment is proposed, as a function of the slope of the linear region of the experimental hat curve only. We apply this analysis to experimental data from which we extract the mechanical quantities: superhelical radius and angle, contact pressure and twisting moment. We compared these values with predictions from previous analyses, when available, and found that they are consistent. In future work, we shall extend the present model to deal with long-range interaction potentials, predict the superhelical radius, and utilize magnetic tweezers experiments to probe DNA-DNA electrostatic interaction. The present paper is a first step towards a mechanically accurate description of bare dsDNA subjected to tensile and torsional loads, a problem relevant to the architecture of DNA in the cell nucleus where proteins come into play. We thank V. Croquette for allowing us to use his unpublished experimental data.
