Microscopic mechanism for experimentally observed anomalous elasticity of DNA in two dimensions.
ABSTRACT By exploring a recent model in which DNA bending elasticity, described by the wormlike chain model, is coupled to basepair denaturation, we demonstrate that small denaturation bubbles lead to anomalies in the flexibility of DNA at the nanometric scale, when confined in two dimensions (2D), as reported in atomic-force microscopy experiments. Our model yields very good fits to experimental data and quantitative predictions that can be tested experimentally. Although such anomalies exist when DNA fluctuates freely in three dimensions (3D), they are too weak to be detected. Interactions between bases in the helical double-stranded DNA are modified by electrostatic adsorption on a 2D substrate, which facilitates local denaturation. This work reconciles the apparent discrepancy between observed 2D and 3D DNA elastic properties and points out that conclusions about the 3D properties of DNA (and its companion proteins and enzymes) do not directly follow from 2D experiments by atomic-force microscopy.
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ABSTRACT: Statistical DNA models available in the literature are often effective models where the base-pair state only (unbroken or broken) is considered. Because of a decrease by a factor of 30 of the effective bending rigidity of a sequence of broken bonds, or bubble, compared to the double stranded state, the inclusion of the molecular conformational degrees of freedom in a more general mesoscopic model is needed. In this paper we do so by presenting a one-dimensional Ising model, which describes the internal base-pair states, coupled to a discrete wormlike chain model describing the chain configurations [J. Palmeri, M. Manghi, and N. Destainville, Phys. Rev. Lett. 99, 088103 (2007)]. This coupled model is exactly solved using a transfer matrix technique that presents an analogy with the path integral treatment of a quantum two-state diatomic molecule. When the chain fluctuations are integrated out, the denaturation transition temperature and width emerge naturally as an explicit function of the model parameters of a well defined Hamiltonian, revealing that the transition is driven by the difference in bending (entropy dominated) free energy between bubble and double-stranded segments. The calculated melting curve (fraction of open base pairs) is in good agreement with the experimental melting profile of poly(dA)-poly(dT) and, by inserting the experimentally known bending rigidities, leads to physically reasonable values for the bare Ising model parameters. Among the thermodynamical quantities explicitly calculated within this model are the internal, structural, and mechanical features of the DNA molecule, such as bubble correlation length and two distinct chain persistence lengths. The predicted variation of the mean-square radius as a function of temperature leads to a coherent explanation for the experimentally observed thermal viscosity transition. Finally, the influence of the DNA strand length is studied in detail, underlining the importance of finite size effects, even for DNA made of several thousand base pairs. Simple limiting formulas, useful for analyzing experiments, are given for the fraction of broken base pairs, Ising and chain correlation functions, effective persistence lengths, and chain mean-square radius, all as a function of temperature and DNA length.Physical Review E 02/2008; 77(1 Pt 1):011913. · 2.31 Impact Factor
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ABSTRACT: Numerical calculations, using Poisson-Boltzmann (PB) and counterion condensation (CC) polyelectrolyte theories, of the electrostatic free energy difference, DeltaGel, between single-stranded (coil) and double-helical DNA have been performed for solutions of NaDNA + NaCl with and without added MgCl2. Calculations have been made for conditions relevant to systems where experimental values of helix coil transition temperature (Tm) and other thermodynamic quantities have been measured. Comparison with experimental data has been possible by invoking values of Tm for solutions containing NaCl salt only. Resulting theoretical values of enthalpy, entropy, and heat capacity (for NaCl salt-containing solutions) and of Tm as a function of NaCl concentration in NaCl + MgCl2 solutions have thus been obtained. Qualitative and, to a large extent, quantitative reproduction of the experimental Tm, DeltaHm, DeltaSm, and DeltaCp values have been found from the results of polyelectrolyte theories. However, the quantitative resemblance of experimental data is considerably better for PB theory as compared to the CC model. Furthermore, some rather implausible qualitative conclusions are obtained within the CC results for DNA melting in NaCl + MgCl2 solutions. Our results argue in favor of the Poisson-Boltzmann theory, as compared to the counterion condensation theory.Biophysical Journal 01/1999; 75(6):3041-56. · 3.67 Impact Factor
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ABSTRACT: The basic features of DNA were elucidated during the half-century following the discovery of the double helix. But it is only during the past decade that researchers have been able to manipulate single molecules of DNA to make direct measurements of its mechanical properties. These studies have illuminated the nature of interactions between DNA and proteins, the constraints within which the cellular machinery operates, and the forces created by DNA-dependent motors.Nature 02/2003; 421(6921):423-7. · 38.60 Impact Factor
Microscopic mechanism for experimentally observed
anomalous elasticity of DNA in 2D
Nicolas Destainville, Manoel Manghi and John Palmeri
Universit´ e de Toulouse; UPS; Laboratoire de Physique Th´ eorique (IRSAMC);
F-31062 Toulouse, France
CNRS; LPT (IRSAMC); F-31062 Toulouse, France
(Dated: March 10, 2009)
By exploring a recent model [Palmeri, J., M. Manghi, and N. Destainville. 2007. Phys. Rev.
Lett. 99:088103] where DNA bending elasticity, described by the wormlike chain model, is coupled
to base-pair denaturation, we demonstrate that small denaturation bubbles lead to anomalies in
the flexibility of DNA at the nanometric scale, when confined in two dimensions (2D), as reported
in atomic force microscopy (AFM) experiments [Wiggins, P. A., et al. 2006. Nature Nanotech.
1:137-141].Our model yields very good fits to experimental data and quantitative predictions
that can be tested experimentally. Although such anomalies exist when DNA fluctuates freely in
three dimensions (3D), they are too weak to be detected. Interactions between bases in the helical
double-stranded DNA are modified by electrostatic adsorption on a 2D substrate, which facilitates
local denaturation. This work reconciles the apparent discrepancy between observed 2D and 3D
DNA elastic properties and points out that conclusions about the 3D properties of DNA (and its
companion proteins and enzymes) do not directly follow from 2D experiments by AFM.
Key words: DNA; denaturation bubble; bending; AFM; wormlike chain
Whereas traditional bulk experiments provide average behaviors of dominant sub-populations, new methods exist
that address DNA mechanical properties at the single-molecule level [1, 2, 3]. Observations by AFM of double-stranded
DNA (dsDNA) adsorbed on a 2D substrate [4, 5] have recently allowed a direct quantification of the distribution, p(θ),
of bending angles θ [6, 7]. This led to the unexpected observation of an over-abundance of large θ , with respect to
the Worm-Like Chain (WLC) model, at very short range (≈ 5 nm, much less than the persistence length ≈ 50 nm).
These observations suggest that, even in the absence of any bending constraints, non-linearities, such as kinks where
2D in air
3D in water
FIG. 1: Sketch of a dsDNA segment solvated in water (left) with its sodium counterion cloud (the phosphate groups of the
DNA backbone are negatively charged); and in air (right), electrostatically adsorbed on a mica substrate forming an ionic
crystal via magnesium ion bridges between the DNA and the negatively charged substrate. The parameters associated with
the hydrogen bonding of bps and the stacking of adjacent bases are therefore significantly modified.
DNA is locally unstacked  or small denaturation bubbles, are excited solely by thermal fluctuations with a high
enough probability to be observable at room temperature (TR= 298.15 K). These findings cast some doubt upon the
adequacy of the WLC model traditionally adopted in 3D . In this respect, Cloutier and Widom  have observed
that short dsDNA, about 100 base-pairs (bp) long, formed looped complexes in 3D with a much higher probability
than expected, which was attributed to partial denaturation . However, these findings have been questioned by
arXiv:0903.1826v1 [cond-mat.stat-mech] 10 Mar 2009
new experiments that pointed out a flaw in the experimental procedure  and showed that short-DNA cyclization
data were accurately fitted by the WLC model, without invoking kinks. A recent study based on flow experiments
draws similar conclusions . These converging elements are supported by all-atom numerical simulations [9, 15]
suggesting that kinks are not excited by thermal fluctuations with any measurable probability in unconstrained DNA
fluctuating freely in solution.
Apart from 2D confinement, what is the difference between both types of experiments? Figure 1 shows a sketch
of DNA fluctuating in solution or adsorbed on a mica surface as in AFM experiments [5, 6, 7]. These experiments
are carried out in air (the solvent is dried) and DNA is electrostatically adsorbed using magnesium ions, forming an
“ionic crystal” with the charged substrate. DNA electrostatics are thus expected to be strongly affected as compared
to DNA in water, hence hydrogen-bonding energies between two complementary bps and stacking energies between
adjacent base aromatic rings are substantially modified.
Recently, we have proposed a solvable model where bending elasticity is intrinsically coupled to bp melting [16, 17]
in contrast to older approaches for which bending is not explicitly included [18, 19]. Single-stranded DNA (ssDNA)
being two orders of magnitude more flexible than dsDNA, this coupling must be taken into account because local
denaturation strongly increases flexibility. Here, we argue that in 2D the modification of the above denaturation
parameters (bonding and stacking energies), due to adsorption, increases the probability of bp opening, which lowers
in turn the bending rigidity. This explanation reconciles the apparent discrepancy between 3D and 2D experiments.
We model dsDNA as a chain of N bps i (1 ≤ i ≤ N) possessing two degrees of freedom [16, 17] : an Ising variable,
σi, set to +1 (resp. −1) when the bp is unbroken (U) (resp. broken, B); in addition to this internal variable, an
external one, the unit vector ti, sets the spatial orientation of the monomer. The Hamiltonian couples explicitly the
κ(σi,σi+1)(1 − ti+1· ti) − J
The bending rigidity of the joint between bps i and i + 1, κ(σi,σi+1), takes different values according to the internal
state of the two neighboring bps. We denote κU ≡ κ(1,1), κB ≡ κ(−1,−1) and κUB ≡ κ(1,−1) = κ(−1,1). The
Ising parameters J and µ have the following physical meanings: J is the destacking energy (energetic cost to unstack
two consecutive aromatic rings); and 2µ is the energy difference per bp between open and closed states.
This discrete WLC model coupled to an Ising one can be completely solved using a transfer matrix approach [16, 17].
Calculating the partition function amounts to solving a spinor eigenvalue problem (formally equivalent to a quantum
rigid rotator). In 3D, the orthogonal eigenstates, denoted by |ˆΨl,m;τ?, are indexed by three “quantum numbers”:
l = 0,1,...,∞ and m = −l,...,l are the usual azimuthal and magnetic quantum numbers associated with the spatial
orientation of tiand τ = ± is related to the “bonding” and “anti-bonding” bp states (as for the one-dimensional Ising
model or the H+
(θ,ϕ) ≡ Ω, the two spherical angles defining t, one gets ?σΩ|ˆΨl,m;τ? = ψl,m(Ω)?σ|l,τ?. The ψl,m(Ω) =√4πYl,m(Ω),
proportional to the spherical harmonics, are the eigenvectors of the pure chain model (i.e. when all κ are set equal).
The eigenvalues λl,τ are degenerate in m and can be expressed in terms of modified Bessel functions of the first kind
Iν(ν = l +1
because if l ?= l?, the matrix element is between states of different rotational symmetry. This is why our coupled
model is not the trivial direct product of both the Ising and discrete WLC models.
The previous exact solution can also be found when the chain is confined to 2D, as already stated by one of us in
Ref. , for example when DNA is adsorbed on a substrate at thermodynamical equilibrium . The spherical angles
(θ,ϕ) become a single polar angle θ ∈ (−π,π]; the spherical harmonics ψl,m(θ,ϕ) become the simpler ψn(θ) = einθ,
with n integer; the two-dimensional analogues of the eigenvalues are denoted by λn,τand the eigenvectors by |n,τ? .
In the model as presented here we do not take into account additional DNA degrees of freedom, such as torsion
or stretching. Although we have recently demonstrated that it is possible to do so in the context of thermal denatu-
ration , the additional mathematical complications of taking them into account in the calculation of the bending
angle distribution would tend to obscure the basic physical mechanism leading to the onset of non-linear effective
bending elasticity and is therefore not warranted here.
2covalent bond). When projecting the eigenstates onto the real space basis |σΩ?, with σ a bp state and
2)  (see Ref.  for the expressions for the |l,τ?). We have ?l,τ?|l,τ? = δττ?, but ?l?,τ?|l,τ? ?= δll?δττ?,
Short-distance chain statistics in 3D and 2D
In order to compute the probability distribution p(ti·ti+r) of finding the polymer with a given relative orientation
between bps i and i + r, we introduce the partial partition function, Z(zi,zi+r), where all degrees of freedom are
integrated out except the projections on the z axis of tiand ti+r, which are set to ziand zi+r(both ∈ [−1,1]):
whereˆP is the transfer matrix and |V ? the boundary vector . The complete calculation from Eq. 2 of p(s) =
4πZ(1,s)/Z, where s ≡ ti· ti+r≡ cosθ, θ is the bending angle between two monomers separated by a distance r,
and Z is the full partition function, is given in the Supplementary Information, B. It uses the decomposition ofˆP on
the eigenbasis |ˆΨl,m;τ?. We have checked that boundary effects are negligible at TRas soon as i is larger than a few
unities. We thus give the final result for p(s) in the limit of long DNA when the internal segment [i,i+r] is far from
both chain ends (i.e. for N → ∞ and i → ∞):
where Pl(s) is a Legendre polynomial . Eq. 3 is a sufficient approximation of Eq. SI.12 for fitting purposes. This
expression reveals the role of infinitely many tangent-tangent correlation lengths, ξp
persistence length, ξp? 150 bp, coincides with the dominant correlation length ξp
The same calculation holds in 2D. We find the following probability distribution (Supplementary Information, C)
× ?V |σ1Ω1?
δ(cosθi− zi)δ(cosθi+r− zi+r)
2l + 1
l,τ= 1/ln(λ0,+/λl,τ). At TR, the
with eigenvalues λn,τ. For the numerical calculation of infinite series such as Eq. 3 or Eq. 4, the sum is performed
up to order 100 (a higher cutoff has been checked not to change numerical values).
n,τ= 1/ln(λ0,+/λn,τ) are also the tangent-tangent correlation lengths associated with 2D eigenmodes |n,τ?
At room temperature, TR, one observes below (see also Fig. 2a,c) that, for θ smaller than a threshold θc, p(s) and
p(θ) coincide with the discrete WLC model probability distribution, pDWLC, which is the simplified version of Eq. 3
or Eq. 4 when no denaturation bubbles appear (formally all κ equal to κU):
2l + 1
in 3D and 2D respectively (dotted lines in Figs. 2a,c), with β = (kBT)−1. In the Gaussian spin-wave approximation
(GSW), βκ ? 1, valid here, the discrete WLC model leads to a quadratic dependence in θ. Indeed, in this case,
κ/r, and pDWLC? pGSW=?βκ/(2πr)exp[−βκθ2/(2r)] in 2D (see Supplementary Information, E). This implies that
the bending rigidity κ and the persistence length ξpare related through ξp= 2βκ in 2D and ξp= βκ in 3D .
2(βκ/r). One ends up with the probability distribution for a single joint of effective bending modulus
the free energy required to bend the polymer by an angle θ is quadratic, F(θ,r) = κθ2/(2r). In this approximation,
We first examine the distribution p(s) ≡ p(ti· ti+r) in 3D. Whereas it is dominated at large r by the largest
persistence length ξp? 150 bp and is well described by the WLC model, this is not true at short r and large θ.
exp. data, r=15
exp. data, r=29
exp. data, r=88
FIG. 2: Theoretical predictions of DNA elastic properties in two and three dimensions. (a) Logarithm of the probability
distribution p(cosθ) = p(s) in 3D (Eq. 3, full lines) for different values of r = 5,15,25 bp (from left to right) compared to the
WLC model (dotted lines). One bp length is a = 0.34 nm. The Ising and elastic parameter values (in units of kBTR) come from
fits to earlier experiments : κU = κUB = 147; κB = 5.54; µ = 1.7977; J = 3.6674. The probability distribution ˜ p(θ) is given
by ˜ p(θ) = sinθ p(cosθ), because ds = sinθ dθ. (b) Logarithm of the probability distributions p(θ) in 2D. Symbols represent
experimental data taken from , whereas the curves are now our best fits, from Eq. 4. The curvilinear distances between
monomers in , namely 5, 10 and 30 nm, correspond respectively to r = 15, 29 and 88 bp. The value κB = 5.54 (in units of
kBTR) comes from Ref.  and κU = 160.82 comes from fitting the r = 88 bp set of data by a pure WLC model, as in 
(because for such large r, the Gaussian character is restored). The remaining parameters (κUB,J,µ) are fitted. One possible
parameter set is (κUB,J,µ) =(20.97,1.3173,1.6685) (see Supplementary Information, D). Dotted line shown the predictions of
the WLC model, for comparison. (c) Logarithm of the probability distribution p(θ) in 2D. Parameter values are coming from
fits (see Panel b), and r = 5,15,25 bp (from top to bottom, full lines). Dotted line shown the predictions of the WLC model
and dashed lines show the same profiles when κB = 0. (d) Average excess chain melting ∆MB(θ) in 2D. Same parameter
values as in Panel b. From left to right, r = 5,15,25 bp. The elasticity is linear until a threshold θc ∝√r, where excessive
bending induces bp melting.
Figure 2a displays the probability density p(s), s = ti· ti+r≡ cosθ, for realistic parameters [16, 17]. At TR, for θ
smaller than a thrshold θc, p(s) coincides with the discrete WLC model distribution, pWLC(s) (Eq. 5), the simplified
version of Eq. 3 when no denaturation bubbles appear. For θ > θc, the plot becomes non-quadratic because of partial
DNA denaturation. The threshold θcis estimated by equating the energetic cost of bending the polymer by an angle
θ in its unmelted state, F(θ,r) = κUθ2
one bp), denoted by ∆GB, which is ∆GB? 17 kBT in 3D . Using this scaling argument, we find
c/(2r), with the free-energy cost of nucleating a single denaturation bubble (of
which gives a good estimate of the observed thresholds (Figure 2a). The anomalies (or non-linearities) appear for
larger and larger values of θ when r grows, and are inexistent in the plots of p(s) as soon as r > 50 bp, i.e. at
length-scales larger than 15 nm, thus recovering standard Gaussian behavior. Indeed, setting θc= π in Eq. 7 yields
the upper limit, rmax? 50 bp, as observed in the plots. This also explains why cyclization experiments with r > 50 bp
are correctly described by the WLC model . For r < 50 bp, this local melting effect is extremely weak, occuring
with a probability?π
is in 2D thermodynamical equilibrium [5, 7]. This is the reason why our statistical mechanical model applies and in
the large N limit, the probability distribution p(θ) is given by Eq. 4. Plots are provided in Figs. 2b,c for realistic
parameter values. At large enough angles, one also sees deviations from the WLC behavior, appearing as soon as
p(θ) ≈ 0.01 rad−1, a now measurable value .
We fit 2D experimental data  in Fig. 2b, using Eq. 4 with κUB, J and µ as fitting parameters (Supplementary
Information, D). The fits are good over the whole θ range. For the best-fit parameter sets, the fraction of melted
bps for unconstrained DNA is then larger than 0.1% at TR, two orders of magnitude higher than in 3D . The
predicted melting temperature, Tm, and transition width, both on the order of 600 K, are also much higher than
their 3D analogues. Despite the high value for Tm in 2D, the large transition width leads, with respect to 3D, to
non-negligible bubble nucleation, even at TR. In other words the loop initiation factor , σ = e−4J0/kBTR≈ 10−2
where J0is the renormalized destacking parameter , is increased by several orders of magnitudes with respect to
3D . The same argument as in 3D leads to rmax? 120 bp in 2D, after modifying ∆GB? 6.6 kBT according to
our fitted parameters. Furthermore, we display in Fig. 2d the average excess of melted bps when ti· ti+r= cosθ is
fixed, as compared to an unconstrained DNA (see Appendix). As anticipated, the deviation from the WLC behavior
at θccoincides with the appearance of melted bps making the polymer more flexible.
θcp(cosθ)sinθdθ ≈ 10−7for r ≥ 5.
The situation is very different when DNA is confined in 2D. It has been demonstrated in experiments that DNA
How can the apparent discrepancy between 2D and 3D parameter values be explained? Not by the fact that the
DNA used in 2D experiments are heteropolymers, whereas the values derived in 3D come from poly(dA)-poly(dT)
homopolymers . Indeed, even for the most robust poly(dG)-poly(dC), Tm ? 360 K in solution. A simple and
straightforward explanation for the discrepancy in parameter values is related to the change in the DNA electrostatic
energy when it is solvated in water (3D) or adsorbed through magnesium (Mg2+) bridges on the mica in a dry
environment.Indeed, it is known that slightly modifying electrostatic interactions (such as by varying the salt
concentration) changes dramatically the denaturation profile of DNA in solution (see e.g. ). The energy required
to break a bp, 2µ, and the energy to destack consecutive bps, 2J, should also be sensitive to the change in the direct
adsorption energy between mica and ds or ssDNA. Strong support for this mechanism comes from the experimental
results of Wiggins et al. themselves . In their Fig. S3, they present the angle distribution and end-to-end distance
statistics for DNA adsorbed on a different-quality mica. Even though the data match to a good approximation those
of their Fig. 3, a detailed analysis of the plots for r = 5 and 7.5 nm leads to the conclusion that the two data sets do
not coincide, even taking into account error bars. This is an experimental indication that the substrate on which DNA
molecules are adsorbed does indeed influence its microscopic parameters. Recent AFM experiments also testified to
a DNA structural modification after adsorption on mica and drying : poly(dG)-poly(dC) proves to shorten its
contour length, supposedly by taking an A-DNA conformation, in contrast to poly(dA)-poly(dT) or plasmid DNA,
both of which keep their B-DNA conformations.
As a result, inferring the parameters µ and J from their 3D analogues is a challenging task. At the present time,
the best strategy is certainly to fit them to experimental data. The above results are confirmed by recent accurate
all-atom molecular dynamics simulations: Mazur has investigated in detail the short-distance angle distribution of
3D DNA and did not find any evidence for the strong deviations from a WLC distribution found experimentally in
Now we discuss in greater detail the role of bubble flexibility, κB, and of cooperativity, J, by comparing our model
to earlier ones. In the kinkable WLC model , kinks of vanishing rigidity can be activated by thermal fluctuations.
This model and ours become physically equivalent in the κB→ 0 limit: a 2 bp denaturation bubble plays the role
of a “kink”, in the sense of a thermally activated local defect without rigidity. Our microscopic vision of a kink
thus differs from Lankas et al.’s local unstacking one , but yields the same short-range mechanical properties.
When κB = 0, the interesting behavior of p(θ) in the denatured region is destroyed: p(θ) becomes flat (Fig. 2c),
as in , and is practically insensitive to r once a kink is nucleated, because a chain segment including a kink has
vanishing rigidity. This is the reason why Wiggins et al. appeal to a different Linear Sub-Elastic Chain (LSEC) model,
with a phenomenological bending energy ELSEC= Λ|θ|, which enables them to satisfactorily fit their experimental
data [7, 28]. In contrast to this LSEC model, our approach proposes a microscopic explanation associated with bubble