Structural ensemble and microscopic elasticity of freely diffusing DNA by direct measurement of fluctuations
Precisely measuring the ensemble of conformers that a macromolecule populates in solution is highly challenging. Thus, it has been difficult to confirm or falsify the predictions of nanometer-scale dynamical modeling. Here, we apply an X-ray interferometry technique to probe the solution structure and fluctuations of B-form DNA on a length scale comparable to a protein-binding site. We determine an extensive set of intrahelix distance distributions between pairs of probes placed at distinct points on the surface of the DNA duplex. The distributions of measured distances reveal the nature and extent of the thermally driven mechanical deformations of the helix. We describe these deformations in terms of elastic constants, as is common for DNA and other polymers. The average solution structure and microscopic elasticity measured by X-ray interferometry are in striking agreement with values derived from DNA-protein crystal structures and measured by force spectroscopy, with one exception. The observed microscopic torsional rigidity of DNA is much lower than is measured by single-molecule twisting experiments, suggesting that torsional rigidity increases when DNA is stretched. Looking forward, molecular-level interferometry can provide a general tool for characterizing solution-phase structural ensembles.
Structural ensemble and microscopic elasticity of freely
diffusing DNA by direct measurement of ﬂuctuations
Xuesong Shi, Daniel Herschlag
, and Pehr A. B. Harbury
Department of Biochemistry, Stanford University, Stanford, CA 94305
by Peter B. Dervan, California Institute of Technology, Pasadena, CA, and approved March 8, 2013 (received for review October 31, 2012)
Precisely measuring the ensemble of conformers that a macromol-
ecule populates in solution is highly challenging. Thus, it has been
difﬁcult to conﬁrm or falsify the predictions of nanometer-scale
dynamical modeling. Here, we apply an X-ray int erferometry
technique to probe the solution structure and ﬂuctuations of
B-form DNA on a length scale comparable to a protein-binding site.
We determine an extensive set of intrahelix distance distributions
between pairs of probes placed at distinct points on the surface
of the DNA duplex. The distributions of measured distances reveal
the nature and extent of the thermally driven mechanical defor-
mations of the helix. We describe these deformations in terms of
elastic constants, as is common for DNA and other polymers. The
average solution structure and microscopic elasticity measured by
X-ray interferometry are in striking agreement with values derived
from DNA–protein crystal structures and measured by force spec-
troscopy, with one exception. The observed microscopic torsional
rigidity of DNA is much lower than is measured by single-molecule
twisting experiments, suggesting that torsional rigidity increases
when DNA is stretched. Looking forward, molecular-level interfer-
ometry can provide a general tool for characterizing solution-phase
bases per helical turn
central lesson from the last 40 y of structural biology is that
proteins and nucleic acids populate multiple conformational
states in solution and that transitions between the states produce
biological function. Despite the importance of such conforma-
tional ﬂuctuations, there is a dearth of tools to quantitatively
measure the ensemble of conformers that is present in solution.
NMR structures are often reported as ensembles, but these
ensembles represent a combination of actual molecular ﬂexibility
and experimental uncertainty. Mo re recently, conformational -
averaged order parameters derived from residual dipolar cou-
pling data have been used to parameterize ensemble models
(1, 2). These models call for testing by an independent experi-
The distances between points in a macromolecule are closely
related to the 3D structure of the macromolecule. This close
relationship is because interpoint distances determine the rela-
tive position of the points in space in a model-free way (allowing
for global rotation, translation, or reﬂection). For a macro-
molecule with a dynamic conformation, distance distributions
between many different pairs of points, in conjunction with
a multibody or elastic model, can deﬁne the macromolecule’s
Thus, in principle, molecular rulers provide the required ex-
perimental information: intramolecular distance distributions.
However, whereas existing rulers are sensitive reporters of or-
dinal change in intramolecular distance, they do not give abso-
lute distances or accurate occupancy distributions when multiple
distinct distances (conformations) coexist. These limitations arise
from averaging of signals over an intrinsic detection time window,
from a complex dependence of the signal on probe and macro-
molecular dynamics in addition to distance, and from nonlinear
and nonunique mapping between the experimental signal and the
underlying distance distribution (3–5). The lack of distance cali-
bration on an absolute scale prevents the quantitative integration
of measurements between different pairs of points and confounds
the comparison of results obtained by different methods with
each other and with computational models.
To address the problem of determining macromolecular struc-
tures in solution, we applied a small-angle X-ray scattering
(SAXS) interferometry technique that provides instantaneous
and high-precision distance information (6, 7). Two gold nano-
crystal probes are attached to a macromolecule, and the mutual
interference in their X-ray scattering is measured (Fig. 1, Left).
Because scattering from bound electrons is fast relative to atomic
motions and because distance is related to the interference pattern
by a Fourier transform, the data directly provide an unaveraged
snapshot of the intramolecular distances between gold probes
that coexist within the solution ensemble (Fig. 1, Right). The dis-
tance distributions are a structural measure of the thermodynamic
landscape of conformational states.
We have applied X-ray interferometry to measure the en-
semble structure of a DNA duplex in solution, building on prior
work that allowed only partial description of its average structure
and conformational ensemble (6). DNA structural excursions
from the canonical Watson–Crick helix are the rule rather than
the exception (8), and these excursions are central to the regu-
lation of biological processes. DNA binding proteins take ad-
vantage of the conformational preferences of different DNA
sequences to enhance recognition speciﬁcity (8–11). Functional
and regulatory events, including the formation of higher-order
chromatin structure, require DNA bending, and the sequence
preferences for bending may pro vide a thermodynamic bias at
the DNA level for controlling gene expression, for the patterning
of nucleosomes on DNA, and possibly for more complex DNA
packing arrangements (refs. 12–16); see also refs. 17 and 18).
Deformation of the double helix is a ubiquitous feature of the
protein–DNA interactions that regulate, replicate, repair, and
pack DNA in cells. Understanding the energetics of DNA de-
formation is therefore of central importance. DNA is generally
modeled as a linear elastic rod, but it has not been possible to
test this directly by observing the nanometer-scale bending
technique, we measured the structural ﬂuctuations of a short
B-form duplex. The results expose a potential nonlinearity of
DNA elasticity and illustrate how to measure the structural
ensemble of a freely diffusing macromolecule.
Author contributions: X.S., D.H., and P.A.B.H. designed research; X.S. performed research;
X.S. analyzed data; and X.S., D.H., and P.A.B.H. wrote the paper.
The authors declare no conﬂict of interest.
This Direct Submission article had a prearranged editor.
To whom correspondence may be addressed. E-mail: firstname.lastname@example.org or
herschl a@st anf ord. edu.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
Published online April 1, 2013 www.pnas.org/cgi/doi/10.1073/pnas.1218830110
We evaluate two models for the DNA structural ensemble that
are based on distinct experimental foundations. The ﬁrst model
treats DNA as a linear elastic rod. Values for the elastic constants
(bending rigidity, stretching rigi dity, twisting rigi dity, and the
coupl ing between them) are taken f rom macros copic single-
molecule stretching and twisting experiments on kilobase-length
DNA fragments. The observed macroscopic elastic behavior is
assumed to extrapolate down to the single base pair level. The
second “knowledge-based” model generalizes from t he large
available databas e of DNA–protein cocrystal structures (19).
The approach involves quantifying six fundamental conforma-
tional variables (twist, tilt, roll, shift, slide and rise; SI Appendix,
Fig. S5B) for each dinucleotide step present in the structural
database, and then ﬁtting the observed values to a normal dis-
tribution. Under the assumption that ﬂuctuations in neighboring
dinucleotide steps are uncorrelated, the normal distributions of
conformational variables are resampled stochastically to gener-
ate collections of model DNA helix coordinates.
Both treatments, the elastic-rod model and the knowledge-
based model, make speciﬁc predictions about the subnanometer
structure and mechanical properties of the DNA duplex. These
predictions have been difﬁcult to test, and there is good reason
to question whether either model is correct. A variety of alter-
native explanations for the microscopic mechanical properties
of DNA (such as occasional large kinks rather than continuous
bends) lead to the same macroscopic wormlike chain behavior
that is observed in single-molecule stretching and twisting ex-
periments, so these experiments cannot deﬁnitively address the
nature of local helix deformations (20). It is also not clear
whether DNA structure in crystals is the same as DNA structure
in solution. Indeed, DNA adopts different structures dependent
on crystal hydration conditions (21). The helix ﬂuctuations infer-
red from naked DNA crystal structures differ considerably from
the ﬂuctuations inferred from DNA–protein cocrystal structures,
and in both cases the ﬂuctuations must be arbitrarily ampliﬁed to
obtain the experimentally measured persistence length of DNA
(22). Direct observation of the structural ensemble of a nanome-
ter-sized helix in solution can overcome these limitations.
Results and Discussion
X-ray interferometry measurements were carried out on DNA
duplexes with gold nanocrystal probes placed at 18 different
pairs of positions. We use the resulting distance distribution data
to determine values for the average DNA helical rise and bases
per turn in solution, and we co mpare them wi th measure ments
made by alternate, less direct experimental techniques. We then
quantitatively evaluate predictions from the linear elastic rod
theory and a knowledge-based theory of DNA elasticity, analyze
the bending and twisting ﬂuctuations obtained from the shape
of the distributions, and compare these results with measure-
ments obtained in force experiments or derived from crystallo-
Distance Distributions by X-Ray Interferometry. To mark speciﬁc
positions on the DNA helix, we coupled gold nanocrystals to the
exocyclic methyl groups of internal T bases through a succini-
midyl 3-(2-pyridyldithio)propionate linker (Materials and Meth-
ods and SI Appendix, Fig. S1). We expected that this linkage
would place the probes on the surface of the DNA helix cylinder,
making them sensitive reporters of helix bending and twisting
ﬂuctuations. [Earlier experiments with gold probes attached
more centrally to the 3′-termini of DNA helices were insen-
sitive to twisting and bending (6).] The distance between a pair of
gold nanocrystal probes depends on the structure of the DNA
double helix, as well as the geometric details of how the probes are
positioned on the helix. Three parameters deﬁne the probe lo-
cation relative to the base pair to which it is attached (Fig. 2): the
displacement along the helix axis of the probe from the base pair
), the azimuthal angular rotation of the probe away from
the short axis of the base pair (θ
), and the radial displacement of
the probe from the center of the helix (D) (6). We globally ﬁt
these parameters to our data, giving the position illustrated in Fig.
2. The large value of D , about 20 Å, supports our prior expecta-
tion that the probe placement would be strongly off axis. The gold
nanocrystal is positioned at the edge of the major groove, with the
van der Waals surface of its thioglucose shell apparently in direct
contact with the phosphodiester backbone. This close packing
likely limits the conformational ﬂexibility of the gold probe,
a feature that enhances our abili ty to obtain high-resolution
conformational information about the DNA itself.
Several lines of evidence suggest that the gold nanocrystal
probes have a negligible inﬂuence on helix structure (SI Appendix,
SI Note 1, Fig. S4, and Table S2). The circular dichroism spectra
of the duplexes is unchanged by labeling, and both labeled and
unlabeled DNA duplex spectra are very different from an A-form
RNA duplex control. In addition, gold labeling alters the melting
temperature of the duplexes by less than 2 °C, and the observed
coupling energy between probe pairs is zero within experimental
error. Finally, if a probe-induced structural perturbation did exist,
distance measurements at progressively increased probe separa-
tions would be fractionally less affected. Consequently, the quality
of the agreement of the measured distance data with the expected
distances from a helical molecule would improve at larger sepa-
rations. No such trend is observed. We also ruled out signiﬁcant
DNA end-fraying effect (23) in our constructs as we found the
results to be independent of how far or close the gold probes are
from the end of the duplex (SI Appendix,Fig.S12).
We systematically measured scattering interference proﬁles
for 18 different gold probe pairs separated by 3–24 base steps
(see Fig. 2D and SI Appendix,TableS1for sequences). The dis-
tribution of center-to-center distances between each probe pair
(Fig. 3) was obtained using procedures outlined schematically
in Fig. 1 and similar in detail to those described previously by
Mathew-Fenn et al. (7) (Fig. 1 and Materials and Methods).
The mean probe separation distance varies systematically with
Fig. 1. Obtaining a probe–probe distance distribution from X-ray interferometry. (Left) DNA duplex labeled with a gold nanocrystal probe on each of the
two strands of DNA (6). After subtracting the scattering signals involving the helix, as indicated by the species above the arrow, the pattern of scattering
interference between the two gold probes is obtained (Center) and Fourier transformed to provide the probability distribution for the center-to-center
distance between the probes (Right) (7). Data shown are for two gold probes separated by 15 base steps within a 26 base pair duplex (see SI Appendix, Table
S1 for sequence).
Shi et al. PNAS
Published online April 1, 2013
the number of intervening base steps. Because the area of each
distribution is normalized to one, a higher peak in the distance
distribution corresponds to a lower variance. Peak heights can
be seen to oscillate up and down with increasing base step sepa-
ration, as expected for a helix with bending and/or twisting motions.
DNA Helix Structure in Solution. Crystal structures of DNA duplexes
have been suggested to provide a reasonable model for approx-
imating DNA structure in solution (19). Nevertheless, the average
helix geometry observed in crystal structures of free DNA differs
from the geometry suggested by biochemical measurements (24).
The interferometry data provide an opportunity to determine
directly the solution helix structure of DNA and to compare this
structure to proposals from prior models.
The mean distance of each observed distance distribution is
plotted in Fig. 4A. Mean distances predicted by the knowledge-
based DNA model are also shown. The dashed line corresponds
to a helix with no adjustable parameters: the rise per base pair (r)
and the number of bases per helical turn (n) are set equal to
literature values from crystal structures of DNA–protein com-
plexes (19) (r = 3.36 Å and n = 10.53 base pairs). Only the three
probe position parameters [the axial probe displacement (axial
the azimuthal probe displacement angle (θ
), and the radial probe
displacement (D)] (6) were ﬁt to the data. The canonical helix
Fig. 2. Gold probe geometry (A–C). Side (A) and top (B) views of a DNA duplex with the gold cores of two nanocrystal probes depicted as spheres. The probes
are shown in the reﬁned position determined from ﬁts to the data (SI Appendix , Table S3A). The top strand of the duplex is drawn in a lighter gray than the
bottom strand. The gold probes linked to the top and bottom strands are colored in yellow and orange, respectively. The ﬁgure shows the probes at a base-
step separation of n = 0, a hypothetical situat ion in which they are attached to the same base pair. Positive (or negative) N values indicate that the yellow gold
sphere on the top strand is displaced relative to the orange gold sphere on the bottom strand by N base steps to the 3ʹ-(or5ʹ-) end of the top strand. Axial
are half of the axial distance and half of the azimuthal angle, respectively, between the two probes at zero base steps. D is the radial displacement of
the probes from the helical axis. The gold core of the probes used herein is 12 Å in diameter (SI Appendix, Fig. S2) and is shown to scale. (C) Atomic model of
thioglucose-passivated nanocrystals coupled to DNA. The nanocrystal coordinates are based on a substructure of the nanocrystal reported in ref. 49 and the
experimental analysis of ref. 50. (D) DNA sequences used in this study. Au nanocrystals were attached at thymines, and these points of attachment are labeled
in red. When aligned to the 3′-end of the top strand, sequences 1a–1d are identical, except for the residues highlighted with a white background in the
Center panel (“Schematics”). Duplexes 2 and 3 have distinct sequences from Duplexes 1a–1d. The tables on the Right (“Base steps investigated”) show the
label positions on the top (or sequence, S) strand and the bottom (or complementary, C) strand; the numbers in the tables refer to the number of base steps
separating the two Au labels. The 11 numbers in red in the top table of the total 18 Au–Au pairs are from sequence 1a or portions of 1b, 1c, and 1d that are
identical in sequence to 1a; the residues in magenta correspond to regions of 1b and 1d that contain the sequence differences. The middle and bottom tables
give the bases steps (in green and blue) for sequence 2 and 3, respectively.
www.pnas.org/cgi/doi/10.1073/pnas.1218830110 Shi et al.
geometry provides a good description of the measured center-to-
center distances between gold nanocrystals. When the two helix
parameters are allowed to vary in addition to the three probe
parameters (ﬁve variables ﬁt to 18 observables), optimal values of
r = 3.55 Å and n = 10.6 base pairs are obtained (solid line, Fig.
4A). The goodness-of-ﬁt decreases steeply as r and n deviate from
the ﬁtted values (SI Appendix,Fig.S3), indicating that the data
provide a strong constraint on the basic geometry of the DNA
helix in solution. The ﬁtted position of the gold nanocrystal
probes is insensitive to the values of helix rise and bases per turn
(SI Appendix, Table S3).
Small deviations of our data from the global ﬁts with uniform
rise and twist values (Fig. 4A) may arise because most of the
measurements used a single DNA sequence (Fig. 2D and SI Ap-
pendix, Table S1), and there may be idiosyncratic properties of
that sequence. Nevertheless, the dominant sequence and the al-
ternate sequences from this study consist of diverse dinucleotide
steps, which is likely to have provided substantial “sequence-
averaging” of the data.
The ﬁtted bases-per-turn value is in excellent agreement with
indirect biochemical measurements (Fig. 4B). The ﬁtted rise
value is somewhat larger than seen in naked DNA crystals and
ﬁbers, most closely matching the value inferred from DNA–
protein cocrystal structures that include outlier dinucleotide step
conformations (Fig. 4B). The interferometry measurements are
consistent with the hypothesis that DNA in solution is better
approximated by protein-bound DNA crystal structures than by
free DNA crystal structures although the differences are modest
and not beyond error.
Microscopic Elasticity of a Freely Diffusing DNA Helix. The shape of
distance distributions can reveal the nature and extent of the
structural ﬂuctuations that deform a macromolecule in solution.
In particular, bending and twisting of a helix produce charac-
teristic oscillations in the width of distance distributions as the
spacing between probes is increased, provided that the probes
are displaced from the helical axis (25) (Fig. 5). Bending-induced
broadening is distinguishable from twisting-induced broadening
because the extrema of the oscillations occur in different probe
arrangements, because the oscillations have different frequen-
cies, and because bending-induced broadening becomes larger
with increased probe separation whereas twisting-induced broad-
ening becomes smaller (Fig. 5).
The experimentally observed variance in distance between gold
nanocrystal probes at different base step separations is plotted
in Fig. 6. The observed oscillation at ∼10.6 base p air intervals
provides direct evidence that nanometer-sized helices undergo
bending ﬂuctuations with a spatial frequency of less th an one
helical repeat and is consistent with models of continuous bend-
ing at the single base pair level. Prior data for the DNA helix in
solution admit the possibility that infrequent localized kinks, such
as those observed in crystal structures of the nucleosome particle
(26), explains its ﬁnite persistence length. However, localized kink
models produce distribution shapes that are inconsistent with the
observed data (SI Appendix, Fig. S11).
The linear elastic rod model (27) and the knowledge-based
model (19) make quantitative predictions for the variance of each
measured distance distribution. These predictions are plotted with
the measured variance data in Fig. 6A. Importantly, there is only
one adjustable parameter in the plotted curves: a constant y-
offset accounting for intrinsic disorder in the position of the gold
nanocrystal probes due to probe heterogeneity or motions of
gold nanocrystals around the linkers. [One potential limitation to
the precision of structural information that can be obtained from
gold nanocrystal probes is the degree of conformational ﬂexi-
bility of the probes with respect to the macromolecule of interest.
The ﬁtted y-intercept values in Fig. 6 are small: less than 5 Å
all of the models. This highly limited probe mobility rules out
explanations for the distance variance that invoke substantial
linker ﬂexibility (28, 29) and will facilitate future high-precision
solution structural measurements.] The predictions of both models
Fig. 3. Experimentally observed distance distributions. The gold probes are
separated by different numbers of positive (Left) or negative (Right) base
steps, as indicated by the colored number labels. The sequences used and
mean distances and variance for each sequence are given in Fig. 2D and SI
Appendix, Table S1.
Fig. 4. Helix geometry in solution from the mean center-to-center distance between gold probes. (A) The experimentally obtained mean probe–probe
separation distances at 18 different base-step separations (circles) are plotted with predicted distances from the knowledge-based model of DNA helix
structure (19). The rise per base and bases per helical turn are set to literature values of 3.36 Å and 10.5 base pairs (19) (black dashed line, χ
= 63) or are ﬁtas
free parameters to give 3.55 Å and 10.6 base pairs (black solid line, χ
= 37). The data are for sequence 1a–1d (red and magenta circles), sequence 2 (green
circles), and sequence 3 (cyan circles), which are shown in Fig. 2D and SI Appendix, Table S1.(B) Helical parameters from ﬁtting o f the interferometry data (red
circle) and from literature measurements (squares and gray/magenta circles). The literature measurements are the r and/or n values from the following: crystal
structures of free DNA (19) (cyan square), crystal structures of DNA–protein complexes (19) (green square), crystal structures of DNA–protein complexes with
outlier conformations removed (19) (blue square), ﬁts to DNA cyclization data (46) (magenta circle), and cleavage periodicity observed in nuclease digestion
experiments (51) (gray circle). The bars on the crystallographic values are SDs of the dinucleotide parameter distributions, and the bars on the experimental
data are 68% conﬁdence intervals.
Shi et al. PNAS
Published online April 1, 2013
are in r easonable agreement with the measured data, although
some points are signiﬁcant outliers.
To investigate how altered helix elasticity within the frame-
work of the linear elastic rod model would affect the predictions,
we varied the stretch modulus (S), the bending persistence length
(B), and the twisting persistence length (C). Threefold changes
in the stretch modulus had negligible effects on the variance
predictions. Conversely, the predictions were very sensitive to the
values of the bending and twisting persistence length. Models
without twisting (Fig. 6C) did not reproduce the variance data
at small base-step separations of the gold probes, and models
without bending (Fig. 6D) gave large deviations from the data at
high base-step separations. The interferometry measurements
thus provide evidence for signiﬁcant ﬂ uctuations via both twisting
and bending on these short length scales. A global search for
elastic constants that optimize the predictions of the linear elastic
rod model yields a bending persistence length of B =55± 10 nm
and a twisting persistence length of C = 20 nm (16–34 nm gave χ
values within 10% of the minimum). These values lead to a sig-
niﬁcantly improved ﬁt to the data (Fig. 6B).
We also evaluated how a proposed cooperative stretching
transition of DNA affects the predictions (4). Addition of a two-
state 0.29-Å stretch (Materials and Methods) to the reparame-
terized linear elastic rod model improves its prediction of the
experimentally determined variances (Fig. 6E). This same model
does a good job of predicting the variance in end-to-end distance
for a series of en d-labeled DNA duplexes that were studied
previously [Fig. 6F; the best-ﬁt value for the stretch of 0.29 Å is
roughly two thirds of a prior estimate (0.42 Å) that did not take
into account variance from bending] (6). Thus, a single model of
the microscopic mechanical properties of DNA can account for
all of the intrahelix distance distributions that have been mea-
sured to date.
How do the microscopic elasticity values measured by X-ray
interferometry compare with previous results? With respect to
bending, the ﬁtted persistence length matches precisely the con-
sensus value of 50–55 nm determined by other methods (30, 31).
However, there is no consensus value for the twisting persistence
length (C) because different experimental techniques give dif-
ferent results. The reported values span a range from 25 to 120
nm. Measurements of twisting diffusion in linear DNA fragments
by time-resolved ﬂuorescence polarization anisotropy (FPA) give
C =25–54 nm (32). Analysis of twist variance in crystal structures
gives C =26–46 nm, depending on whether naked DNA or DNA–
protein complexes are used, and on how outlier dinucleotide step
conformations are eliminated (19, 33). Analysis of the circulari-
zation kinetics and topoisomer distributions of short DNA frag-
ments produces C estimates between 58 and 80 nm (34, 35)
whereas topoisomer distribution analysis of longer DNA frag-
ments, where the bending strain is smaller, gives C = 49 nm (36).
Finally, single-molecule torque measurements on kilobase-length
DNA fragments under tension give C = 100–120 nm (37–39), at
least twice the magnitude of the other estimates. The distance
distributions measured by X-ray interferometry indi cate th at
short DNA helices in solution undergo extensive microscopic
twisting ﬂuctuations, with a twist persistence length of only
∼20 nm (Fig. 6B) that lies at the short end of the reported range.
The much higher torsional rigidity observed in single-molecule
torque experiments may be a consequence of DNA stretching,
which is required in those experiments to distinguish twist
from writhe (39). The implied strong dependence of torsional
rigidity on stretching and bending [as observed in topoisomer
distribution analysis and by FPA (40)] suggests a need for addi-
tional experimental tests and a description of DNA elasticity that
incorporates nonlinear effects.
An alternate ensemble-modeling approach is to run molecular
dynamics simulations constrained by experimental data. For ex-
ample, a model ensemble of the Dickerson DNA dodecamer has
been proposed, based on combining extensive NMR measure-
ments and large angle X-ray scattering data with molecular dy-
namics calculations (41). Excluding the terminal base pairs, this
ensemble gives a long twist persistence length (93 nm), falling
near the value from single-molecule twist experiments. On the
other hand, the bending persistence length from this ensemble is
extremely low (7.2 nm), ∼sevenfold smaller than the consensus
value (50–55 nm).
Fig. 5. Signatures of helix bending and twisting in the variance of simulated distance distributions. (A) Bending ﬂuctuations give rise to peaks and valleys in
the distance distribution variance with increasing probe separation. Variance maxima occur when two probes are located on the same side of the helix, and
variance minima occur when the probes are located on opposite sides of the helix. The bending-induced variance oscillates once per helical turn. (B) Twisting
ﬂuctuations also give rise to peaks and valleys in distribution variance. Variance maxima occur when two probes are at roughly right angles to each other, and
variance minima occur when the probes are either on the same side of the helix or on opposi te sides of the helix. The twisting-induced variance oscillates
twice per helical turn. For both bending and twisting, the magnitude of the oscillations increases steeply as the probes are positioned further away from the
helix axis (compare D = 10 Å in blue versus D = 20 Å in orange). Note: Although drawn in two dimensions, the positions of the extrema in the twisting variance
also depend on the vertical separation between the two probes (i.e., twisting in three dimensions).
www.pnas.org/cgi/doi/10.1073/pnas.1218830110 Shi et al.
Despite the striking agreement of the interferometry data with
predictions from current models, discrepancies exist between the
data and even the best model predictions (Figs. 4 and 6). These
differences may be sequence-speciﬁc effects or reﬂect properties
of DNA that are not currently included in the models. One
possibility is the existence of cooperative conformational changes
that extend over multiple base pairs. A known example is runs
of four or more consecutive A-bases, which form an A-tract helix
structure that differs from helix structures with three or fewer
consecutive A bases (42). X-ray interferometry can distinguish
these structural differences and promises to elucidate other se-
quence-speciﬁc helical properties, as well as the inﬂuence of pro-
teins and other ligands on DNA conformation in solution. Also,
whereas the model of the ensemble of DNA conformations pre-
sented above likely captures the majority of occupied regions of
DNA’s energy landscape under nonperturbing solution condi-
tions, it does not include higher energy and very rarely sampled
states that are also of functional importance in biology. Such
high-energy states include sh arply kinked conformatio ns t hat
allow for circularization of short DNA fragments and likely
participate in chromatin packing, and helices with non-Watson–
Crick base pairs or bases ﬂipped out for enzymatic modiﬁcation
and repair (43, 44). Nonetheless, these high-energy states are
rarely sampled and do not contribute measurably to the ensemble
distance distributions at room temperature.
Conclusions and Implications
X-ray interferometry offers a powerful complement to other
solution approaches, such as NMR spectroscopy, optical rulers,
and single-molecule mechanical probing, by providing calibrated
and unambiguous atomic-scale distance information. In the case
of DNA, there was previously no reliable way to measure bending
and twisting rigidity at the microscopic length scale of less than
∼30 base pairs (45). The measured geometric values are of high
precision and are directly comparable with distances within
single structures dete rmined by diffraction from crystals. With
sufﬁcient probe sets, the method can be used to quantitatively
and precisely determine the structural ensemble of a macromol-
ecule in solution.
Although this study was not designed to investigate the se-
qu ence dependence of DNA elasticity, our data can be compared
with two nearest neighbor elasticity models (19, 46). The data do
not provide support for either model, and systematic variation of
duplex sequences will be required to determine the scale and
nature of such effects.
As a means to measure the mechanical properties of macro-
molecules, X-ray interferometry has unique advantages. It is not
restricted to regular polymeric materials, and the method natu-
rally applies to globular proteins and structured RNAs. It operates
under nonperturbing conditions, for example with no mechanical
load and in the presence of physiological salt concentrations,
Fig. 6. Observed pattern of probe–probe distance variation and the predictions of different mechanical models. (A) The experimentally obtained distance-
distribution variances (circles) are plotted together with predicted values based on the linear elastic rod model (blue line, χ
= 8.4) and the knowledge-based
model (brown line, χ
= 6.0). (B) Variance predictions of a reparameterized linear elastic rod model (black line, χ
= 5.4). The bending and twisting rigidity
were optimized together with the ﬁve probe and helical parameters (SI Appendix, Table S3B) so as to minimize the χ
of a ﬁt against both the mean and
variance data. The optimized bending persistence length is 55 ± 10 nm, and the optimized twisting persistence length is 20 nm (16–34 nm give χ
differ by less than 10%). (C) Variance predictions of the reparameterized linear elastic rod model (black) with no twisting ﬂuctuations (yellow; χ
= 14). (D)
Variance predictions of the reparameterized linear elastic rod model (black) with no bending ﬂuctuations (yellow; χ
= 29). (E) Variance predictions of the
reparameterized linear elastic rod model without (black) and with a 0.29 Å per base pair cooperative stretching transition (red line, χ
= 4.9). (F) End-to-end
distance variance of DNA duplexes measured previously (6) (circles) and variance predictions of the reparameterized linear elastic rod model with (red line,
= 3.8) and without (black line, χ
= 20) a cooperative stretching transition. The y-intercept values ﬁt to the data are 4.2, 0.5, 0.4, 0.0, and 3.7 Å
for the linear elastic rod model (blue line in A), the knowledge-based model (brown line in A), the reparameterized linear elastic rod model (B), and the
reparameterized linear elast ic rod model with a cooperative stretch ( E and F). These small intercepts, which approximate the contributi on to the variance
from ﬂexibility of the Au nanocrystal attachment to the DNA, suggest that there is little residual motion of the probe. The experimental data are from
sequences 1a–1d (red and magenta), sequence 2 (green), and sequence 3 (cyan), which are shown in Fig. 2D and SI Appendix, Table S1.
Shi et al. PNAS
Published online April 1, 2013
requiring only controls to ensure that the attached nanocrystals
do not alter the underlying conformational ensemble. Finally,
the interferometry technique can be used on multiple length
scales. We studied a nanometer-sized object here, but we could
equally well have measured distances in a large macromolecular
complex using bigger nanocrystal probes.
X-ray interferometry should be particularly useful for studying
intrinsically dynamic nucleic acids such as functional RNAs,
protein ensembles such as those of allosteric enzymes, molten
globules and natively unstructured polypeptides, and molecular
machines that operate via multistep reaction cycles. It also pro-
vides an experimental means to assess the strengths and limi-
tations of molecular dynamics simulations, as distance distribu-
tions can be readily extracted from both interferometry data and
computational trajectories and directly compared. Such com-
parisons will be powerful in further deciphering and deﬁning
macromolecular ensembles and dynamics and their underlying
Materials and Methods
Materials. Gold nanocrystals were synthesized and puriﬁed as described
previously (6). SPDP [succinimidyl 3-(2-pyridyldithio)propionate] was pur-
chased from Thermo Scientiﬁc. DNA oligonucleotides were synthesized on
an ABI 393 DNA synthesizer and puriﬁed by Poly-Pak cartridge (Glen Re-
search) followed by anion exchange HPLC (Dionex DNAPac 100, 10 mM to
1.5 M NaCl in 20 mM sodium borate buffer, pH 7.9). The procedure for
preparing labeled DNA–gold constructs was similar to that in ref. 6. Oligo-
nucleotides with amino-modiﬁed thymine (40 nmol; Amino-Modiﬁer C2 dT,
Glen Research) in 140 μL of 0.1 M sodium borate buffer, pH 7.9, were reacted
with 20 μL of an SPDP solution (1 mg/10 μL in DMSO) at 37 °C for 30 min. The
reaction was continued for another 30 min after addition of a second 20-μL
aliquot of the SPDP solution. Following ethanol precipitation to remove
excess SPDP, the SPDP-modiﬁed oligonucleotides were treated with 100 mM
DTT at 70 °C for 30 min in 50 mM Tris·HCl, pH 9.0, to reduce the internal
disulﬁde bonds, and were precipitated again to remove excess DTT. The
thiol-containing oligonucleotides were incubated for 2 h with a ﬁvefold
molar excess of gold nanocrystals in 20 mM Tris·HCl, pH 9.0. Gold-coupled
oligonucleotides were puriﬁed by anion exchange HPLC (DNAPac 100:
10 mM to 1.5 M NaCl in 20 mM ammonium acetate, pH 5.6), hybridized with
the appropriate complementary strand for 30 min at room temperature, and
repuriﬁed by a second anion exchange HPLC step (performed as above ) to
obtain the pure double-labeled duplex. The samples were desalted and
concentrated using centrifugal ﬁlter devices (10-kDa cutoff, Millipore). Final
sample purity was assessed by HPLC. The overall yield for the double-labeled
duplex is 15–25% of the initial puriﬁed, unlabeled oligonucleotide.
SAXS Measurements and Data Processing. Small-angle X-ray scattering mea-
surements were carried out at beamline 4–2 of the Stanford Synchrotron
Radiation Lightsource (SSRL) using a sample-to-detector distance of 1.7 m or
1.1 m. A buffer of 150 mM NaCl, 70 mM Tris·HCl, pH 8.0, with 10 mM sodiu m
ascorbate was used for all experiments. Data were acquired and analyzed
following previously published procedures (7) with two modiﬁcations de-
scribed below. X-ray scattering proﬁ les were measured for six samples: the
double gold-labeled DNA duplex (AB), the single gold-labeled DNA duplexes
(A and B), the unlabeled DNA duplex (U), the gold nanocrystals alone (Au),
and buffer alone (Buf).
The probe-probe scattering interference proﬁle, I
, was calculated as: I
. C denotes relative concentration,
I denotes scattering intensity, and the subscripts indicate sample identit y as
deﬁned above. This expression differs from ref. 7 in that it omits the term
that was used previously to subtract scattering contributions from
free gold nanocrystals. No free nanocrystals ( <1%) could be detected in the
samples reported here. The scaling coefﬁcients C
, and C
chosen to minimize the function T:
S < 0:06
ðSÞ × S
S > 0:06
ðSÞ × S
ðSÞ × S
ðDÞ < 0
where S is the magnitude of the scattering vector, U
is the differ ence radial
Patterson de ﬁ ned in ref. 7, and D is distance. Minimizing T ensures that the
sinusoidal oscillations in I
(S) × S average to zero and penalizes negative
features in the difference radial Patterson (negative features are unphysical,
because the electron density of the macromolecule exceeds the solvent
electron density). This expression for T weights low-S data more heavily
than the scaling target function used in ref. 7 where the beamline
setup all owed data co llecti on t o h igher scattering angles. T he minimum/
maximumvaluesofthescalingcoefﬁcients ﬁt to the 18 datasets are: 0.54/
1.09 for C
, 0.91/1.2 for C
,and−0.05/0.08 for C
. Final probe–pr ob e
distance distributions were obtained by decomposing I
, into bas is in ter-
ference p ro ﬁles corresponding to discrete center-to-center separation
distances between probes. The decomposition was performed using a maxi-
mum entropy algorithm.
Because some of the observed distance distributions were asymmetrical
(Fig. 3), we did not approximate distributions with single Gaussian curves as
in ref. 6 to calculate mean distances and variances. Instead, the mean and
variance of each distribution were calculated respectively as: <d> = ∑P
, where P
is the relative probability of
ﬁnding a probe pair at a distance d
tances within ±2.5 SD of a Gaussian curve ﬁt to the central feature of the dis-
tribution. The same procedure was used to calculate the mean and variance of
probe–probe distance distributions predicted by DNA mechanical models.
Predictions from Models. Model probe–probe distance distributions were
obtained by constructing a virtual DNA chain of 10
base pairs as described
in ref. 29. Each base pair in the chain was built upon the previous base pair
using values of the dinucleotide parameters (twist, tilt, roll, shift, slide and
rise; SI Appendix, Fig. S5B) selected by random Guassian sampling of the
eigenvectors of a diagonalized covaration matrix (47). For the knowledge-
based model, the covariation matrix was compiled from the structural
parameters observed in a select set of DNA–protein crystal structures (19).
Parameters for each of the 16 possible dinucleotide steps were treated in-
dependently. The construction algorithm used a randomly generated DNA
sequence in which each of the 16 dinucleotide steps occurred with equal
frequency. Following ref. 22, the covariation matrices were ampliﬁed by
a factor of (0.85)
so that the bending persistence length of the modeled
DNA would be 50 nm. The twist-persistence length of the DNA generated by
the knowledge-based model was 39 nm. For the linear elastic rod model, the
shift and slide parameters were set to zero. A force matrix based on the
assumed stretch modulus, torsional persistence length, bending persistence
length, and twist-stretch coupling constant was constructed. The force ma-
trix was then inverted and multiplied by k
T at 298 K to give a matrix of
covariation in the twist, tilt, roll, and rise parameters (tilt and roll shared
a common angular probability distribution in the elastic rod model). To
simulate cooperative stretching, base pairs could switch between two states
with rise values that were 0.14 Å less than or more than the mean rise value.
(The ±0.14 Å value was obtained from a ﬁt of the cooperative elastic rod
model to the data in ref. 6.) The likelihood that a base pair would switch
state relative to its predecessor was set to 1 in 80 as in ref 6.
Each set of six dinucleotide parameters deﬁnes a transformation matrix
relating the local coordinate frame of the previous base pair to that of its
successor. The center position of a gold probe in the coordinate frame of the
labeled base pair is calculated as [D × cos(θ
), D × sin(θ
] (Fig. 2 and SI
Appendix, Table S3). Application of the appropriate transf ormation matrices
gives the probe center position in the coordinate frame of adjacent base
pairs, so that center-to-center distances between two probes can be com-
puted. The predicted ensemble of distances for a probe pair separated by
N base steps was generated by moving a pair of virtual gold particles at
positions i and i + N down the chain (∼10
samples). The mean and variance
of the modeled distributions were determined as described above for the
experimentally measured distributions. SI Appendix, SI Note 2 provides the
detailed generating information for each model.
Fitting the Gold Probe Position, Helical Rise, and Base Pairs per Turn. Two helical
parameters (the average rise per base and the average number of bases per
helical turn) were varied in addition to the three probe position parameters.
For each choice of parameter values, a virtual DNA chain of 10
base pairs was
constructed as described above and used to compute distributions of probe–
probe distance for base-step separations between −35 and 35. A χ
quantifying the goodness-of-ﬁt between the mean values of the model dis-
tributions and the observed distributions was then computed. The set of
parameters that minimized the χ
statistic were identiﬁed by a numerical
search using MATLAB’s fminsearch algorithm.
Fitting the Bending and Twisting Persistence Length. Fits were performed as
described above, with variation of three additional parameters: B, the DNA
bending persistence length; C, the DNA twisting persistence length; and e,
the variance attributed to gold probe heterogeneity and linker ﬂexibility.
The optimal parameters were deﬁned as those that minimized a sum of the
www.pnas.org/cgi/doi/10.1073/pnas.1218830110 Shi et al.
statistic quantifying the goodness-of-ﬁ t between the means of the model
and observed distributions plus seven times the χ
statistic quantifying the
goodness-of-ﬁt between the variances of the model and observed dis-
tributions. The factor of seven roughly equalizes the magnitudes of the two
sums. Optimal parameter values were identiﬁed by a numerical search
using the genetic algorithm toolbox in MATLAB (48).
ACKNOWLEDGMENTS. We thank H. Tsuruta, T . Matsui, and T. Weiss at
Beamline 4-2 of the Stanford Synchrotron Radiation Lightsource (SSRL) for
technical support in synchrotron small-angle X-ray scattering experiments,
members of the D.H. and P.A.B.H. laboratories, R. Das, and R. S. Math ew for
helpful discussions, and R. Sengupta and J. Caldwell for comments on the
manuscript. This work was supported by National Institutes of Health Grants
DP-OD000429-01 (to P.A.B.H.) and GM49243 (to D.H.).
1. Zhang Q, Stelzer AC, Fisher CK, Al-Hashimi HM (2007) Visualizing spatially correlated
dynamics that directs RNA conformational transitions. Nature 450(7173):1263–1267.
2. Lange OF, et al. (2008) Recognition dynamics up to microseconds revealed from an
RDC-derived ubiquitin ensemble in solution. Science 320(5882):1471–1475.
3. Jeschke G (2012) DEER distance measurements on proteins. Annu Rev Phys Chem 63:
4. Kalinin S, Sisamakis E, Magennis SW, Felekyan S, Seidel CAM (2010) On the origin of
broadening of single-molecule FRET efﬁciency distributions beyond shot noise limits.
J Phys Chem B 114(18):6197–6206.
5. Kruschel D, Zagrovic B (2009) Conformational averaging in structural biology: issues,
challenges and computational solutions. Mol Biosyst 5(12):1606–1616.
6. Mathew-Fenn RS, Das R, Harbury PAB (2008) Remeasuring the double helix. Science
7. Mathew-Fenn RS, Das R, Silverman JA, Walker PA, Harbury PAB (2008) A molecular
ruler for measuring quantitative distance distributions. PLoS ONE 3(10):e3229.
8. Rohs R, et al. (2010) Origins of speciﬁcity in protein-DNA recognition. Annu Rev
9. Rohs R, et al. (2009) The role of DNA shape in protein-DNA recognition. Nature
10. Hizver J, Rozenberg H, Frolow F, Rabinovich D, Shakked Z (2001) DNA bending by an
adenine—thymine tract and its role in gene regulation. Proc Natl Acad Sci USA 98(15):
11. Otwinowski Z, et al. (1988) Crystal structure of trp repressor/operator complex at
atomic resolution. Nature 335(6188):321–329.
12. Drew HR, Travers AA (1985) DNA bending and its relation to nucleosome positioning.
J Mol Biol 186(4):773–790.
13. Kaplan N, et al. (2009) The DNA-encoded nucleosome organization of a eukaryotic
genome. Nature 458(7236):362–366.
14. Morozov AV, et al. (2009) Using DNA mechanics to predict in vitro nucleosome
positions and formation energies. Nucleic Acids Res 37(14):4707–4722.
15. Rippe K, von Hippel PH, Langowski J (1995) Action at a distance: DNA-looping and
initiation of transcription. Trends Biochem Sci 20(12):500–506.
16. Robertson CA, Nash HA (1988) Bending of the bacteriophage lambda attachment site
by Escherichia coli integration host factor. J Biol Chem 263(8):3554–3557.
17. Zhang Y, et al. (2010) Evidence against a genomic code for nucleosome positioning.
Nat Struct Mol Biol 17(8):920
18. Zhang Y, et al. (2009) Intrinsic histone-DNA interactions are not the major determinant
of nucleosome positions in vivo. Nat Struct Mol Biol 16(8):847–852.
19. Olson WK, Gorin AA, Lu XJ, Hock LM, Zhurkin VB (1998) DNA sequence-dependent
deformability deduced from protein-DNA crystal complexes. Proc Natl Acad Sci USA
20. Wiggins PA, Nelson PC (2006) Generalized theory of semiﬂexible polymers. Phys Rev E
Stat Nonlin Soft Matter Phys 73(3 Pt 1):031906.
21. Shakked Z, Guerstein-Guzikevich G, Eisenstein M, Frolow F, Rabinovich D (1989) The
conformation of the DNA double helix in the crystal is dependent on its environment.
22. Olson WK, Colasanti AV, Czapla L, Zheng G (2008) Insights into the sequence-
dependent macromolecular properties of DNA from base-pair level modeling. Coarse-
Graining of Condensed Phase and Biomolecular Systems, ed Voth GA (Crc Press-Taylor
& Francis Group, Boca Raton, FL), pp 205–223.
23. Nikolova EN, Al-Hashimi HM (2009) Preparation, resonance assignment, and
preliminary dynamics characterization of residue speciﬁc 13C/15N-labeled elongated
DNA for the study of sequence-directed dynamics by NMR. J Biomol NMR 45(1-2):
24. Tereshko V, Subirana JA (1999) Inﬂuence of packing interactions on the average
conformation of B-DNA in crystalline structures. Acta Crystallogr D Biol Crystallogr
25. Becker NB, Everaers R (2009) Comment on “Remeasuring the double helix”. Science
325(5940):538–, author reply 538.
26. Richmond TJ, Davey CA (2003) The structure of DNA in the nucleosome core. Nature
27. Gore J, et al. (2006) DNA overwinds when stretched. Nature 442(7104):836–839.
28. Mazur AK (2009) Analysis of accordion DNA stretching revealed by the gold cluster
ruler. Phys Rev E Stat Nonlin Soft Matter Phys 80(1 Pt 1):010901.
29. Zheng GH, Czapla L, Srinivasan AR, Olson WK (2010) How stiff is DNA? Phys Chem
Chem Phys 12(6):1399–1406.
30. Bustamante C, Bryant Z, Smith SB (2003) Ten years of tension: Single-molecule DNA
mechanics. Nature 421(6921):423–427.
31. Hagerman PJ (1988) Flexibility of DNA. Annu Rev Biophys Biophys Chem 17:265–286.
32. Fujimoto BS, Brewood GP, Schurr JM (2006) Torsional rigidities of weakly strained
DNAs. Biophys J 91(11):4166–4179.
33. Matsumoto A, Olson WK (2002) Sequence-dependent motions of DNA: A normal
mode analysis at the base-pair level. Biophys J 83(1):22
34. Horowitz DS, Wang JC (1984) Torsional rigidity of DNA and length dependence of
the free energy of DNA supercoiling. J Mol Biol 173(1):75–91.
35. Shore D, Baldwin RL (1983) Energetics of DNA twisting. I. Relation between twist and
cyclization probability. J Mol Biol 170(4):957–981.
36. Taylor WH, Hagerman PJ (1990) Application of the method of phage T4 DNA ligase-
catalyzed ring-closure to the study of DNA structure. II. NaCl-dependence of DNA
ﬂexibility and helical repeat. J Mol Biol 212(2):363–376.
37. Lipfert J, Kerssemakers JWJ, Jager T, Dekker NH (2010) Magnetic torque tweezers:
Measuring torsional stiffness in DNA and RecA-DNA ﬁlaments. Nat Methods 7(12):
38. Bryant Z, et al. (2003) Structural transitions and elasticity from torque measurements
on DNA. Nature 424(6946):338–341.
39. Moroz JD, Nelson P (1997) Torsional directed walks, entropic elasticity, and DNA twist
stiffness. Proc Natl Acad Sci USA 94(26):14418–14422.
40. Heath PJ, Clendenning JB, Fujimoto BS, Schurr JM (1996) Effect of bending strain on
the torsion elastic constant of DNA. J Mol Biol 260(5):718–730.
41. Schwieters CD, Clore GM (2007) A physical picture of atomic motions within the
Dickerson DNA dodecamer in solution derived from joint ensemble reﬁnement
against NMR and large-angle X-ray scattering data. Biochemistry 46(5):1152–1166.
42. Haran TE, Mohanty U (2009) The unique structure of A-tracts and intrinsic DNA
bending. Q Rev Biophys 42(1):41–81.
43. Vafabakhsh R, Ha T (2012) Extreme bendability of DNA less than 100 base pairs long
revealed by single-molecule cyclization. Science 337(6098):1097–1101.
44. Cheng XD (1995) Structure and function of DNA methyltransferases. Annu Rev
Biophys Biomol Struct 24:293–318.
45. Fujimoto BS, Schurr JM (2005) Can reliable torsion elastic constants be determined
from FPA data on 24 and 27 base-pair DNAs? Biophys Chem 116(1):41–55.
46. Geggier S, Vologodskii A (2010) Sequence dependence of DNA bending rigidity. Proc
Natl Acad Sci USA 107(35):15421–15426.
47. Czapla L, Swigon D, Olson WK (2006) Sequence-dependent effects in the cyclization
of short DNA. J Chem Theory Comput 2(3):685–695.
48. Goldberg DE (1989) Genetic Algorithms in Search, Optimization and Machine Learning
(Addison Wesley Publishing Company, Boston, MA).
49. Jadzinsky PD, Calero G, Ackerson CJ, Bushnell DA, Kornberg RD (2007) Structure of a
thiol monolayer-protected gold nanoparticle at 1.1 A resolution. Science 318(5849):
50. Dass A (2009) Mass spectrometric identiﬁcation of Au68(SR)34 molecular gold
nanoclusters with 34-electron shell closing. J Am Chem Soc 131(33):11666–11667.
51. Rhodes D, Klug A (1980) Helical periodicity of DNA determined by enzyme digestion.
Shi et al. PNAS
Published online April 1, 2013