Structural ensemble and microscopic elasticity of freely
diffusing DNA by direct measurement of fluctuations
Xuesong Shi, Daniel Herschlag1, and Pehr A. B. Harbury1
Department of Biochemistry, Stanford University, Stanford, CA 94305
Edited†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 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 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
Au-SAXS|bending rigidity|twisting rigidity|persistence length|
bases per helical turn
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. More 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
central lesson from the last 40 y of structural biology is that
proteins and nucleic acids populate multiple conformational
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 provide 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
and twisting of the helix. Using an X-ray interferometry
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.
1To whom correspondence may be addressed. E-mail: email@example.com or
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
| Published online April 1, 2013www.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 rigidity, twisting rigidity, and the
coupling between them) are taken from macroscopic 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 the large
available database 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 compare them with measurements
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
doublehelix,aswellasthegeometricdetailsofhow the probes are
positioned on the helix. Three parameters define the probe lo-
cationrelative to the base pair towhichit is attached(Fig.2): the
displacement along the helix axis of the probe from the base pair
(axial0), the azimuthal angular rotation of the probe away from
the short axis of the base pair (θ0), and the radial displacement of
the probe from the center of the helix (D) (6). We globally fit
theseparameters to ourdata,giving theposition illustratedin 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 ability 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, Table S1 for 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
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).
Obtaining a probe–probe distance distribution from X-ray interferometry. (Left) DNA duplex labeled with a gold nanocrystal probe on each of the
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 (axial0),
the azimuthal probe displacement angle (θ0), and the radial probe
displacement (D)] (6) were fit to the data. The canonical helix
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 situation 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- (or 5ʹ-) end of the top strand. Axial0
and θ0are 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.
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
| www.pnas.org/cgi/doi/10.1073/pnas.1218830110Shi 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 pair intervals
provides direct evidence that nanometer-sized helices undergo
bending fluctuations with a spatial frequency of less than 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
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 Å2for
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
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.
Experimentally observed distance distributions. The gold probes are
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, χ2= 63) or are fit as
free parameters to give 3.55 Å and 10.6 base pairs (black solid line, χ2= 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 of 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.
Helix geometry in solution from the mean center-to-center distance between gold probes. (A) The experimentally obtained mean probe–probe
Shi et al. PNAS
| Published online April 1, 2013
are in reasonable 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 fluctuations 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 χ2
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 end-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
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 indicate that
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).
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 opposite 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).
Signatures of helix bending and twisting in the variance of simulated distance distributions. (A) Bending fluctuations give rise to peaks and valleys in
| 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 sharply kinked conformations that
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 determined 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-
quence 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,
distribution variances (circles) are plotted together with predicted values based on the linear elastic rod model (blue line, χ2= 8.4) and the knowledge-based
model (brown line, χ2= 6.0). (B) Variance predictions of a reparameterized linear elastic rod model (black line, χ2= 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 χ2of 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 χ2values that
differ by less than 10%). (C) Variance predictions of the reparameterized linear elastic rod model (black) with no twisting fluctuations (yellow; χ2= 14). (D)
Variance predictions of the reparameterized linear elastic rod model (black) with no bending fluctuations (yellow; χ2= 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, χ2= 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,
χ2= 3.8) and without (black line, χ2= 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 Å2, respectively,
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 elastic rod model with a cooperative stretch (E and F). These small intercepts, which approximate the contribution 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.
Observed pattern of probe–probe distance variation and the predictions of different mechanical models. (A) The experimentally obtained distance-
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| 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 sodium
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 profiles 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, IΔ, was calculated as: IΔ=
IAB− CA+B× (IA+ IB) + CU× IU− CBuf× IBuf. C denotes relative concentration,
I denotes scattering intensity, and the subscripts indicate sample identity as
defined above. This expression differs from ref. 7 in that it omits the term
CAu× IAuthat 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 CU, CA+B, and CBuf were
chosen to minimize the function T:
T ¼∑S<0:06½IΔðSÞ×S?2×S=Sminþ ∑S>0:06½IΔðSÞ×S?2
where S is the magnitude of the scattering vector, UΔis the difference radial
Patterson defined in ref. 7, and D is distance. Minimizing T ensures that the
sinusoidal oscillations in IΔ(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 allowed data collection to higher scattering angles. The minimum/
maximum values of the scaling coefficients fit to the 18 datasets are: 0.54/
1.09 for CU, 0.91/1.2 for CA+B, and −0.05/0.08 for CBuf. Final probe–probe
distance distributions were obtained by decomposing IΔ, into basis inter-
ference profiles 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> = ∑Pi×
di/∑Piand σ2= ∑Pi× (di− <d>)2/∑Pi, where Piis the relative probability of
finding a probe pair at a distance di. Each summation was restricted to dis-
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 106base 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)−1so 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 kBT 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(θ0), D × sin(θ0), axial0] (Fig. 2 and SI
Appendix, Table S3). Application of the appropriate transformation 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 (∼106samples). 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 106base pairs was
constructed as described above and used to compute distributions of probe–
probe distance for base-step separations between −35 and 35. A χ2statistic
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 χ2statistic 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.
χ2statistic quantifying the goodness-of-fit between the means of the model Download full-text
and observed distributions plus seven times the χ2statistic 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
χ2sums. 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. Mathew 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.).
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| Published online April 1, 2013