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Doublestranded RNA under force and torque:
Similarities to and striking differences from
doublestranded DNA
Jan Lipfert
a,b
, Gary M. Skinner
a,1
, Johannes M. Keegstra
a,2
, Toivo Hensgens
a
, Tessa Jager
a
, David Dulin
a
,
Mariana Köber
a
, Zhongbo Yu
a
, Serge P. Donkers
a
, FangChieh Chou
c
, Rhiju Das
c,d
, and Nynke H. Dekker
a,3
a
Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands;
b
Department of
Physics, Nanosystems Initiative Munich, and Center for NanoScience, Ludwig Maximilians University Munich, 80799 Munich, Germany; and
c
Departments of
Biochemistry and
d
Physics, Stanford University, Stanford, CA 94305
Edited by Ignacio Tinoco Jr., University of California, Berkeley, CA, and approved September 17, 2014 (received for review April 18, 2014)
RNA plays myriad roles in the transmission and regulation of genetic
information that are fundamentally constrained by its mechanical
properties, including the elasticity and conformational transitions
of the doublestranded (dsRNA) form. Although doublestranded
DNA (dsDNA) mechanics have been dissected with exquisite pre
cision, much less is known about dsRNA. Here we present a com
prehensive characterization of dsRNA under external forces and
torques using magnetic tweezers. We find that dsRNA has a force–
torque phase diagram similar to that of dsDNA, including plecto
neme formation, melting of the double helix induced by torque,
a highly overwound state termed “PRNA,”and a highly under
wound, lefthanded state denoted “LRNA.”Beyond these similari
ties, our experiments reveal two unexpected behaviors of dsRNA:
Unlike dsDNA, dsRNA shortens upon overwinding, and its charac
teristic transition rate at the plectonemic buckling transition is two
orders of magnitude slower than for dsDNA. Our results challenge
current models of nucleic acid mechanics, provide a baseline for mod
eling RNAs in biological contexts, and pave the way for new classes
of magnetic tweezers experiments to dissect the role of twist and
torque for RNA–protein interactions at the singlemolecule level.
RNA

nucleic acids

magnetic tweezers

force

torque
RNAs are central to many biological processes. In addition to
wellcharacterized roles as messenger, transfer, ribosomal,
viral, and spliceosomal RNA, RNA molecules have more recently
discovered functions including enzymatic activity, gene silencing,
and sensing of metabolites. In many of these contexts, structures
rich in doublestranded RNA (dsRNA) helices encounter me
chanical strains; examples include the packaging of dsRNA viral
genomes into capsids, deformations of the ribosome during
translation (1, 2), and more generally conformational changes of
functional RNAs while folding or due to interactions with proteins
(3, 4). In addition, RNA is emerging as a material for engineered
nanostructures both in vitro (5) and in vivo (6). A quantitative
understanding of these processes requires accurate knowledge of
the elastic properties and conformational transitions of RNA
under forces and torques.
For doublestranded DNA (dsDNA), the mechanical proper
ties and structural transitions under forces and torques have been
mapped out rigorously (7–10). Its elastic responses to bending,
stretching, and twisting deformations of the standard Bform
helix (Fig. 1 Aand B), characterized by the bending persistence
length A, the stretch stiffness S, the torsional persistence length
C, and the twist–stretch coupling D, have been accurately de
termined using singlemolecule manipulation techniques (SI
Appendix, Table S1 and Materials and Methods). In addition,
singlemolecule techniques have provided a comprehensive view
of the force–torque phase diagram of dsDNA (7, 9, 11). Knowl
edge of the elastic constants and conformational transitions of
dsDNA has had a tremendous impact and set the stage for
implementing, modeling, and interpreting numerous experiments
involving DNA (7, 8, 10), its interactions with proteins (12, 13) and
other binding partners, its behavior in confined environments, and
its assembly into engineered nanostructures (14).
In contrast, much less is known about dsRNA, despite its
overall structural similarity. Like DNA, RNA can form right
handed double helices. In contrast to DNA, RNA forms an Aform
helix with a radius of ∼1.2 nm and a length increase per base
pair of ∼2.8 Å, ∼20% wider and shorter than Bform dsDNA (Fig.
1A). Although recent singlemolecule stretching experiments using
torsionally unconstrained dsRNA have revealed its bending persis
tence length (15, 16), stretch modulus (16), and an overstretching
transition (16, 17), its response to torsional strains and structural
transitions under forces and torques is unknown. This dearth of in
formation on dsRNA is partially due to the relative difficulty, com
pared with dsDNA, of assembling RNA constructs suitable for
singlemolecule force and torque measurements. Here we use single
molecule magnetic tweezers (MTs) measurements on fully torsion
ally constrained dsRNA molecules to provide a comprehensive view
of dsRNA mechanics that includes its complete elastic response, its
force–torque phase diagram, and its dynamics of loop formation.
Results
Torsionally Constrained dsRNA Constructs for Magnetic Tweezers. We
constructed fully doublestranded RNA constructs with multiple
Significance
RNA, like DNA, can form double helices held together by the
pairing of complementary bases, and such helices are ubiquitous
in functional RNAs. Here we apply external forces and torques
to individual doublestranded RNA molecules to determine the
mechanical properties and conformational transitions of these
fundamental biological building blocks. For small forces and
torques, RNA helices behave like elastic rods, and we have de
termined their bending, stretching, and twisting stiffness. Sur
prisingly, we find that RNA shortens when it is overwound,
whereas DNA lengthens. Finally, we twist RNA until it buckles
and forms a loop, and find the timescale of this transition to be
much slower for RNA compared with DNA, suggesting un
expected differences in their flexibilities on short length scales.
Author contributions: J.L., G.M.S., and N.H.D. designed research; J.L., G.M.S., J.M.K., T.H.,
T.J., D.D., M.K., Z.Y., S.P.D., F.C.C., and R.D. performed research; J.L., J.M.K., T.H., F.C.C.,
and R.D. analyzed data; and J.L., F.C.C., R.D., and N.H.D. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
Freely available online through the PNAS open access option.
1
Present address: Illumina UK, Little Chesterford, Essex CB10 1XL, United Kingdom.
2
Present address: Systems Biology, FOM Institute for Atomic and Molecular Physics, 1098
XG Amsterdam, The Netherlands.
3
To whom correspondence should be addressed. Email: n.h.dekker@tudelft.nl.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
1073/pnas.1407197111//DCSupplemental.
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attachment points at both ends suitable for MTs torque mea
surements by annealing two complementary single strands that
carry multiple biotin or digoxigenin labels at their respective 5′
ends (Fig. 1 Cand Dand Materials and Methods). The function
alized singlestranded constructs were generated by carrying out
initial in vitro transcription reactions that incorporated labeled
nucleotides and stalled at a missing fourth nucleotide (Fig. 1 Cand
D). After purification, transcription reactions were restarted and
completed in the presence of all four unlabeled nucleotides. The
final annealed 4.2kbp dsRNA constructs can be tethered between
an antidigoxigenin–coated flow cell surface and streptavidin
coated magnetic beads for manipulation in the MTs (Fig. 1E).
Force–Extension Response of dsRNA. Using the ability of MTs to
exert precisely calibrated stretching forces (18, 19) (Materials and
Methods and SI Appendix, Fig. S1), we first probed the force–
extension response of dsRNA. The stretching behavior of
torsionally relaxed dsRNA at low forces (F<5 pN) is well
described by the (inextensible) wormlike chain (WLC) model
(20, 21) (SI Appendix, Fig. S2A). From fits of the WLC model, we
determined the contour length L
C
=1.15 ±0.02 μm and the
bending persistence length A
RNA
=57 ±2 nm in the presence of
100 mM monovalent salt (SI Appendix, Fig. S2A), in good
agreement with the expected length (1.16 μm, assuming 0.28 nm
per bp) (22, 23) and previous singlemolecule stretching experi
ments (15, 16). A
RNA
decreases with increasing ionic strength
(16) (SI Appendix, Fig. S1), in a manner welldescribed by models
that partition it into an electrostatic and a saltindependent
component (SI Appendix, Fig. S1K). Taking into account the salt
dependence, A
RNA
is consistently ∼20% larger than A
DNA
at the
same ionic strength (SI Appendix, Fig. S1).
Stretching dsRNA at forces >10 pN, we observed elastic
stretching that can be fit by the extensible WLC model (21, 24) up
to ∼40 pN (SI Appendix, Fig. S2B) and an overstretching transition
for torsionally unconstrained molecules (SI Appendix, Fig. S2C), in
agreement with previous singlemolecule studies (16, 17). From
fits of the extensible WLC model, we found S
RNA
=350 ±100 pN,
about threefold lower than S
DNA
(SI Appendix,Fig.S1Gand Table
S1). Our value for the S
RNA
is in reasonable agreement with, al
though slightly lower than, the value of S
RNA
∼500 pN determined
in singlemolecule optical tweezers measurements (25), possibly
due to subtle differences between magnetic and optical tweezers
experiments. For torsionally unconstrained molecules, the over
stretching transition is marked by a rapid increase in extension to
1.8 ±0.1 times the crystallographic length over a narrow force
range at F=54 ±5pN(SI Appendix, Fig. S2C). In contrast, using
our torsionally constrained dsRNA, we observed enthalpic
stretching beyond the contour length but no sharp overstretching
transition up to F=75 pN (SI Appendix, Fig. S2D). The increased
resistance to overstretching for torsionally constrained dsRNA
compared with torsionally unconstrained dsRNA is qualitatively
similar to the behavior of dsDNA (26–28) (SI Appendix, Fig. S1 H
and I). The dependence of the overstretching transition for
dsRNA on torsional constraint and on salt (SI Appendix, Fig. S2 C
and D) suggests that it might involve melting as well as a tran
sition to a previously unidentified conformation that we name
“SRNA,”in analogy to SDNA (SI Appendix, Fig. S1).
Twist Response of dsRNA. We used the ability of MTs to control
the rotation of the magnetic beads (18) to map out the response
of dsRNA upon over and underwinding at constant stretching
forces. Starting with a torsionally relaxed molecule (corresponding
to zero turns in Fig. 2), the tether extension remains initially ap
proximately constant upon overwinding (corresponding to in
creasing linking number) until the molecule reaches a buckling
point (Fig. 2A, dashed lines and SI Appendix,Fig.S3). Further
overwinding beyond the buckling point leads to a rapid linear de
crease of the tether extension with an increasing number of turns,
due to the formation of plectonemes. The critical supercoiling
density σ
buck
for buckling increases with stretching force and agrees
within experimental error with the values found for DNA and with
a mechanical model originally developed for supercoiled DNA (9)
(Fig. 2Band SI Appendix,Materials and Methods). The decrease in
extension per added turn in the plectonemic regime provides
a measure for the size of the plectonemes and decreases with in
creasing stretching force (Fig. 2C). The extension vs. turns slopes
for dsRNA are within experimental error of those for dsDNA, and
are in approximate agreement with the mechanical model for
supercoiling (Fig. 2C). Underwinding the dsRNA tether at
stretching forces F<1 pN gives rise to a buckling response similar
to what is observed upon overwinding and the formation of neg
atively supercoiled plectonemes. In contrast, for F>1 pN, the
over and underwinding response is asymmetric and the tether
extension remains approximately constant upon underwinding
(Fig. 2A), likely due to melting of the double helix, as has been
observed for DNA (29) (SI Appendix, Fig. S3 Kand L).
If unwinding at F>1 pN is continued for several hundred
turns, we eventually observe another structural transition marked
by an abrupt change in the extension vs. turns response at a
supercoiling density of σ∼–1.9 (Fig. 2D). We term this previously
unidentified highly underwound and lefthanded RNA confor
mation with a helicity of –12.6 bp per turn “LRNA,”in analogy to
what has been observed for highly underwound DNA (11) (SI
Appendix,Fig.S3L). We note that the helicity and elongation that
Fig. 1. Construction of a torsionally constrained doublestranded RNA for
magnetic tweezers measurements. (A) Comparison of Aform dsRNA [Protein
Data Bank (PDB) ID code 1RNA (57)] and Bform dsDNA [PDB ID code 2BNA
(58)]. (B) Cartoon of the elastic deformations of dsRNA: be nding, stretching, and
twisting. (C) Schematic of the protocol to generate doublestranded RNA
molecules with multiple attachment points at both ends. Initial transcription
reactions incorporate multiple biotinylated adenosine (green circles) or digoxi
genated uracil (yellow squares) bases and stall at a fourth nucleotide. After
purification, transcription reactions are restarted and complete the 4.2kbp
transcripts. In the final step, the purified RNA strands are annealedto yi eld dsRNA
with chemical modifications at each end. (D) Schematic of the two DNA tem
plates used to generate dsRNA with multiple labels at both ends. (E)Cartoonof
a magnetic tweezers experiment on dsRNA (not to scale). A streptavidincoated
magnetic bead is tethered to an antidigoxigenin–coated surface by a dsRNA
molecule with multiple attachment points at both ends. A surfaceattached ref
erence bead is tracked simultaneously for drift correction. Permanent magnets
above the flow cell are used to exert a stretching force Fand to control the ro
tation of the magnetic bead via its preferred axis m
0
. N, north pole; S, south pole.
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we observe for LRNA under torsional constraint are similar to
what has been proposed for the NMR solution structure of a short
(6bp) GCrich dsRNA fragment in 6 M monovalent salt (30).
However, further investigation is necessary to elucidate structural
details of torsionally strained lefthanded dsRNA.
Finally, for F>5 pN, dsRNA ceases to undergo a buckling
transition even upon overwinding (Fig. 2A, top curve). We pro
pose that dsRNA undergoes a transition to a highly overwound
conformation termed “PRNA”under these conditions, in ana
logy to experimentally observed PDNA (31) and in line with
modeling predictions based on molecular dynamics simulations of
dsRNA (32).
To further quantify the torsional response of dsRNA, we carried
out magnetic torque tweezers (33–35) measurements that directly
monitor the torque response of the nucleic acid tether upon over
and underwinding by tracking the rotation angle about the tether
axis and using a modified magnet geometry compared to conven
tional magnetic tweezers (Fig. 3 Aand Band SI Appendix,Fig.S4).
Starting from a torsionally relaxed molecule (corresponding to zero
turns), we initially observe a linear response of the torque to over
and underwinding (Fig. 3C). Upon overwinding beyond the linear
response regime, the torque saturates when the molecule under
goes the buckling transition (for F<5 pN; marked by a concomi
tant rapid decrease in the tether extension; Fig. 3D) or the AtoP–
form transition (for F>5pN;atacriticaltorqueΓ
AtoP
=38.3 ±
2pN·nm). We determined the values of the postbuckling torque
Γ
buck
as a function of stretching force from the torque plateaus in
theplectonemicregime(Fig.3E). Similar to σ
buck
,Γ
buck
for dsRNA
agrees within experimental error with the values determined for
dsDNA and with a simple mechanical model (Fig. 3E). Immedi
ately before the torque assumes the plateau value Γ
buck
,weobserve
atorque“overshoot,”qualitatively similar to what has been re
cently reported for dsDNA (35, 36) (Fig. 3C,Inset). Upon under
winding, the torque saturates when the molecule buckles and forms
negative plectonemes (for F<1 pN; again marked by a rapid
decrease in tether extension) or melts (for F>1 pN; at a melting
torque of −11 ±1pN·nm, independent of stretching force).
Fig. 2. Response of dsRNA to changes in linking number at various stretch
ing forces. (A) Rotation–extension curves for dsRNA at different stretching
forces (0.25, 0.5, 1, 2, 4, and 5.5 pN, from dark to light). The top axis shows the
supercoiling density, σ=ΔLk/Lk
0
(SI Appendix,Materials and Methods).
Dashed lines denote the buckling points at positive turns, and solid lines
denote linear fits to the extension in the plectonemic region. (B) Critical
supercoiling density for buckling as a function of applied forc e for dsRNA and
dsDNA. A simple mechanical model for supercoiling (8) predicts the right
trend (dashed line), whereas a refined model (9) provides a good fit to the
dsRNA data with the torsional stiffness of the plectonemic state (P) set
to 23 ±3 nm (solid line). (C) Slope of the rotation–extension curves in the
plectonemic regime at σ>0 for dsRNA and dsDNA. The 16kbp dsDNA data are
from ref 59. The simple mechanical mode again predicts the right trend (dashed
line), whereas the refined model provides an approximate fit to the dsRNA data
with P =20 ±3 nm (solid line). Data points in Band Care means and SEM of at
least five independent measurements. (D) Rotation–extension curves for dsRNA
out to large negative σat F=0.5, 2, 3, 6, and 7.5 pN (dark to light). Solid lines
indicate unwinding; dashed lines indicate subsequent rewinding. All data pre
sented were obtained in the presence of 100 mM NaCl.
Fig. 3. Torque response of dsRNA at various stretching forces. (A)Schematicof
a magnetic torque tweezers (MTTs) measurement on dsRNA. The MTTs are
a variant of MTs that enables the measurement of torque. (B) Principle of
torque measurements in MTTs. After overwinding (or underwinding) the
dsRNA tether by Nturns, the dsRNA exerts a restoring torque on the bead that
leads to a shift in the equilibrium angular position from θ
0
to θ
N
. This shift can
be directly converted to torque (SI Appendix,Fig.S4). (C) Rotation–torque
curves for 4.2kbp dsRNA at F=0.5, 1, 3, and 6.5 pN (dark to light). Gray lines
correspond to fits to the torque plateaus to determine buckling and melting
torques. Colored lines are linear fits to determine the torsional stiffness. (Inset)
Additional data for F=3pN.(D)Rotation–extension curves corresponding to
the measurements in C. Solid lines indicate linear fits in the plectonemic regime.
(E) Buckling torques as a function of applied stretching force for dsRNA and
dsDNA, determined from the plateaus in the rotation–torque data at positive
turns. The data points at 6.5 pN (triangles) correspond to the critical torques for
PRNA and PDNA formation. The prediction of a simple mechanical model for
supercoiling (8) captures the right trend (dashed line), whereas a refined model
(9) provides a good fit to the dsRNA data with the torsional stiffness of the
plectonemic state set to P =21.6 ±2 nm (solid line). (F) Effective twist persis
tence length Cfor dsRNA and dsDNA as a function of Fdetermined from
linear fits of the torque vs. applied turns data in the elastic twist regime. The
lines are fits of the Moroz–Nelson model (37), with the high force data (F>
2.5 pN; solid lines) yielding limiting values of C
RNA
=100 ±2 nm and C
DNA
=
109 ±4 nm. Data points for dsRNA in Eand Fare means and SEM of at least
five independent measurements; data for 7.9kbp DNA are from ref. 34. (G)
Phase diagram for dsRNA as a function of applied force and torque. Red
points connected by solid lines correspond to transitions directly measured in
this work. Dashed lines correspond to putative transition regions that have
not been directly observed. A, Aform dsRNA; −scA and +scA, negatively and
positively supercoiled Aform dsRNA, respectively. LRNA, PRNA, and SRNA
denote the alternative dsRNA conformations discussed in the main text.
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We determined the effective twist persistence length C
RNA
from the slopes in the linear torque–response regime, where
thetorqueafterNturns is 2π·N·k
B
T·C
RNA
/L
C
(where k
B
is
Boltzmann’s constant and Tis the absolute temperature; Fig. 3C,
solid colored lines). C
RNA
increases with increasing force and
is 99 ±5nmatF=6.5 pN. Compared with dsDNA, C
RNA
is
similar to but slightly lower than C
DNA
, and both quantities
exhibit similar force dependence, in qualitative agreement
with a model valid in the high force limit (37) (Fig. 3Fand SI
Appendix,Materials and Methods). Combining the results from
stretching and torque measurements at different forces, we de
lineate the phase diagram for dsRNA as a function of applied
force and torque (Fig. 3G).
Twist–Stretch Coupling. The linear elastic rod model has a fourth
parameter, D, that describes the coupling between twist and
stretch. We measured the twist–stretch coupling for dsRNA by
monitoring changes in the extension upon over and under
winding while holding the molecule at constant stretching forces
that are large enough to suppress bending and writhe fluctua
tions (38, 39) (Fig. 4A). We found that for small deformations (in
the range –0.02 <σ<0.025, which excludes the melting, buck
ling, and AtoP–form transitions) dsRNA shortens upon over
winding, with a slope of (dΔL/dN)
RNA
=–0.85 ±0.04 nm per
turn, independent of stretching force in the range F=4–8pN
(Fig. 4 Band C). This is in stark contrast to dsDNA, which
we observed to lengthen upon overwinding by (dΔL/dN )
DNA
=
+0.44 ±0.1 nm per turn (Fig. 4 Band C), in good agreement
with previous measurements (38–41). Our measurements suggest
that dsRNA has a positive twist–stretch coupling equal to D
RNA
=
–S
RNA
·(dΔL/dN)
RNA
/(2π·k
B
T)=+11.5 ±3.3 (assuming S
RNA
=350
pN; SI Appendix,Materials and Methods), in contrast to the negative
twist–stretch coupling of dsDNA (38–41), D
DNA
=–17 ±5.
Dynamics at the Buckling Transition. Next, we investigated the dy
namics at the buckling transition. When a dsRNA was twisted
close to the critical supercoiling density, we observed jumps in the
extension traces, corresponding to transitions between the pre
and postbuckling states (Fig. 5A). Recording extension traces at
a fixed number of applied turns, the population of the post
buckling state increases whereas the population of the prebuck
ling state decreases with an increasing number of applied turns
(Fig. 5A). After selecting a threshold to separate the pre and
postbuckling states (SI Appendix,Fig.S5A–D), the pre and
postbuckling populations and dwell time distributions can be
quantified. The dependence of the postbuckling population on
the number of applied turns is welldescribed by a twostate
model (42) (Fig. 5Band SI Appendix,Materials and Methods)from
which we determined the number of turns converted from twist to
writhe during the buckling transition ΔN
b
∼4 turns (SI Appendix,
Fig. S5L). The dwell times in the pre and postbuckling state are
exponentially distributed (SI Appendix,Fig.S5E–G), and their
mean residence times depend exponentially on the number of
applied turns (Fig. 5C). We determined the overall characteristic
buckling times τ
buck
, that is, the dwell times at the point where the
pre and postbuckling states are equally populated, from fits of
the exponential dependence of the mean residence times on the
number of applied turns (Fig. 5Cand SI Appendix,Materials and
Methods). τ
buck
increases with increasing salt concentration and
stretching force (Fig. 5E). The force dependence of τ
buck
is well
described by an exponential model (solid lines in Fig. 5E), τ
buck
=
τ
buck,0
·exp(d·F/k
B
T); from the fit we obtain the buckling time at
zero force τ
buck,0
=13 and 52 ms and the distance to the transition
state along the reaction coordinate d=5.1 and 5.5 nm for the 100
and 320 mM monovalent salt data, respectively.
Interestingly, comparing τ
buck
for dsRNA with dsDNA of
similar length under otherwise identical conditions (Fig. 5 Dand
E), we found that the buckling dynamics of dsRNA are much
slower than those of dsDNA, with the characteristic buckling
times differing by at least two orders of magnitude. For example,
we found τ
buck
=10.1 ±3.7 s for dsRNA compared with ∼0.05 s
for dsDNA at F=4 pN and 320 mM salt (Fig. 5E).
Discussion
Our experiments are consistent with dsRNA behaving as a linear
elastic rod for small deformations from the Aform helix, and
allow us to empirically determine all four elastic constants of
the model: A,S,C, and D(SI Appendix, Table S1). To go beyond
the isotropic rod model, toward a microscopic interpretation
of the results, we describe a “knowledgebased”base pairlevel
model that considers the six basestep parameters slide, shift,
rise, twist, roll, and tilt (SI Appendix, Fig. S6 and Materials and
Methods; a full description of modeling for a blind prediction
challenge is given in ref. 43). Average values and elastic cou
plings of the basestep parameters for dsRNA and dsDNA from
a database of nucleic acid crystal structures are used in a Monte
Carlo protocol to simulate stretching and twisting experiments (SI
Appendix,Materials and Methods). This base pairlevel model
correctly predicts the bending persistence length for dsRNA to be
slightly larger than for dsDNA, S
RNA
to be at least a factor of two
smaller than S
DNA
, and Cto be of similar magnitude for dsRNA
and dsDNA (SI Appendix, Table S2). The significant difference in
stretch modulus Sbetween dsRNA and dsDNA can be explained
from the “springlike”path of the RNA base pairs’center axis,
compared with dsDNA (SI Appendix,Fig.S6B). Beyond the
agreement with experiment in terms of ratios of dsRNA and
dsDNA properties, the absolute values of A,S,andCall fall within
a factor of two of our experimental results for both molecules.
Whereas the values for A,S, and Care fairly similar for
dsRNA and dsDNA, our experiments revealed an unexpected
difference in the sign of the twist–stretch coupling Dfor dsRNA
and dsDNA. The twist–stretch coupling has important biological
Fig. 4. Doublestranded RNA has a positive twist–stretch coupling. (A)Time
traces of the extension of a dsRNA tether held at F=7 pN and underwound by
−6 or overwound by 12turns. Raw traces (120 Hz) are in red and filtered data
(10 Hz) are in gray. The data demonstrate that dsRNA shortens when over
wound. (B) Changes in tether extension upon over and underwinding at F=7
pN of a 4.2kbp dsRNA and a 3.4kbp dsDNA tether. Linear fits to the data
(lines) indicate that the dsDNA lengthens by ∼0.5 nm per turn, whereas the
dsRNA shortens by ∼0.8 nm per turn upon overwinding. Symbols denote the
mean and standard deviation for four measurements on the same molecule.
(C) Slopes upon overwinding of dsRNA and dsDNA tethers as a function of F
(mean and SEM of at least four molecules in TE +100 mM NaCl buffer). Data
of Lionnet et al. (38) are shown as a black line with the uncertainty indicated
in gray; data from Gore et al. (39) are shown as a black square. The red line is
the average over all dsRNA data. (D) Models of oppositely twisting 50bp
segments of dsDNA (Left)anddsRNA(Right) under 0 and 40 pN stretching
forces, derived from base pairlevel models consistent with experimental
measurements (SI Appendix, Table S6 and Materials and Methods). The or
ange bars represent the long axis of the terminal base pair.
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implications, such as for how mutations affect binding sites, be
cause a base pair deletion or insertion changes not only the
length but also the twist of the target sequence, changes that
need to be compensated by distortions of the nucleic acid upon
protein binding (39). Nevertheless, accounting for the twist–
stretch coupling Din a model of nucleic acid elasticity appears
to be challenging. Previous elastic models originally developed
for dsDNA (44, 45) predict a positive twist–stretch coupling
for dsRNA, in agreement with our measurements for D
RNA
al
though at odds with the results for dsDNA (SI Appendix,Mate
rials and Methods). In contrast, elastic models that consider a stiff
backbone wrapped around a softer core give negative Dpre
dictions for both dsRNA and dsDNA (39, 46). Likewise, the base
pairlevel Monte Carlo model yields a negative twist–stretch cou
pling for both dsDNA and dsRNA, disagreeing with the positive
sign we observe for D
RNA
(SI Appendix,TableS2), although we
note that relatively modest changes to the basestep parameters
can reproduce the experimentally observed value for D
RNA
(Fig. 4Dand SI Appendix,Materials and Methods). Interestingly,
an allatom, implicitsolvent model of dsDNA homopolymers
found Aform dsDNA to unwind upon stretching whereas Bform
dsDNA overwound when stretched close to its equilibrium con
formation (47). Although these simulation results are in qualitative
agreement with our findings for Aform dsRNA and Bform
dsDNA, their simulation predicts un and overwinding, respectively,
by ∼3° per 0.1 nm, which corresponds to values of jDj∼50, namely
a factor of three to five larger in magnitude than the experimen
tally observed values for D
RNA
and D
DNA
. In summary, a complete
microscopic understanding of the twist–stretch coupling for both
dsRNA and dsDNA may require higherresolution (allatom,
explicitsolvent) models and novel experimental methods.
A second surprising contrast between dsRNA and dsDNA is
the much slower buckling dynamics for dsRNA. The two orders
of magnitude difference in τ
buck
is particularly astonishing, be
cause the parameters that characterize the end points of the
buckling transitions and the difference between them, such as
σ
buck
(Fig. 2B), Γ
buck
(Fig. 3E), the extension jump (SI Appendix,
Fig. S5I), and ΔN
b
(SI Appendix,Fig.S5L), are all similar (within
at most 20–30% relative difference) for dsRNA and dsDNA.
Several models that describe the buckling transition in an elastic
rod framework (characterized by Aand C) find reasonable agree
ment between experimental results for dsDNA and the parame
ters that characterize the end points of the buckling transition (42,
48–50). In contrast, there is currently no fully quantitative model
for the buckling dynamics. A recent effort to model the timescale
of the buckling transition for dsDNA found submillisecond
buckling times, much faster than what is experimentally observed,
suggesting that the viscous drag of the micrometersized beads or
particles used in the experiments might considerably slow down
the observed buckling dynamics for dsDNA (48).
The observed difference in τ
buck
suggests that the transition
state and energy barrier for buckling are different for dsRNA and
dsDNA. We speculate that because the transition state might
involve sharp local bending of the helix (on a length scale of ∼5
nm, suggested by the fit to the force dependence; Fig. 5E), the
observed difference might possibly be due to high flexibility of
dsDNA on short length scales, which would lower the energetic
cost of creating sharp transient bends. An anomalous flexibility of
dsDNA on short length scales is hotly debated (51), and has been
suggested by different experiments, including cyclization assays in
bulk using ligase (52) or at the singlemolecule level using FRET
(53), smallangle Xray scattering measurements on goldlabeled
samples (54), and atomic force microscopy imaging of surface
immobilized DNA (55), even though the evidence remains con
troversial (51). If the observed difference in τ
buck
between
dsDNA and dsRNA is indeed due to an anomalous flexibility of
dsDNA on short length scales, a clear prediction is that similar
experiments for dsRNA should fail to observe a corresponding
level of flexibility. In addition, this striking, unpredicted differ
ence between dsDNA and dsRNA again exposes a critical gap in
current modeling of nucleic acids.
In conclusion, we have probed the elastic responses and struc
tural transitions of dsRNA under applied forces and torques. We
find the bending and twist persistence lengths and the force–tor
que phase diagram of dsRNA to be similar to dsDNA and the
stretch modulus of dsRNA to be threefold lower than that of
dsDNA, in agreement with base pairlevel model predictions.
Surprisingly, however, we observed dsRNA to have a positive
twist–stretch coupling, in agreement with naïve expectations but in
contrast to dsDNA and to base pairlevel modeling. In addition,
we observe a striking difference of the buckling dynamics for
dsRNA, for which the characteristic bucklingtransition time is two
orders of magnitude slower than that of dsDNA. Our results
provide a benchmark and challenge for quantitative models of
nucleic acid mechanics and a comprehensive experimental foun
dation for modeling complex RNAs in vitro and in vivo. In addi
tion, we envision our assay to enable a new class of quantitative
singlemolecule experiments to probe the proposed roles of twist
and torque in RNA–protein interactions and processing (4, 56).
Materials and Methods
See SI Appendix,Materials and Methods for details. In brief, the double
stranded RNA constructs for magnetic tweezers experiments were generated
by annealing two 4,218kb complementary singlestranded RNA molecules that
carry multiple biotin or digoxigenin labels at their respective 5′ends (Fig. 1C).
The product of the annealing reaction is a 4,218bp (55.6% GC content) fully
doublestranded RNA construct with multiple biotin labels at one end and
Fig. 5. Slow buckling transition for dsRNA. (A) Time traces of the extension
of a 4.2kbp dsRNA tether for varying numbers of applied turns (indicated on
the far right) at the buckling transition for F=2 pN in 320 mM NaCl. (Right)
Extension histograms (in gray) fitted by double Gaussians (brown lines). Raw
data were acquired at 120 Hz (gray) and data were filtered at 20 Hz (red).
(Inset) Schematic of the buckling transition. (B) Fraction of the time spent in
the postbuckling state vs. applied turns for the data in Aand fit of a twostate
model (black line; SI Appendix,Materials and Methods). (C) Mean residence
times in the pre and postbuckling state vs. applied turns for the data in A
and fits of an exponential model (lines; SI Appendix,Materials and Meth
ods). (D) Extension vs. time traces for dsRNA (red) and dsDNA (blue) both at
F=4 pN in TE buffer with 320 mM NaCl added. Note the different timescales
for dsRNA and dsDNA. (E) Characteristic buckling times for 4.2kbp dsRNA in
TE buffer with 100 mM (red points) and 320 mM (orange points) NaCl added
(mean and SEM of at least four independent molecules). Solid lines are fits of
an exponential model. Measurements with 3.4kbp dsDNA tethers in 320 mM
NaCl at F=4 pN yielded characteristic buckling times of ∼50 ms (horizontal
dashed line); however, this value represents only an upper limit, because our
time resolution for these fast transitions is biased by theacquisition frequency
of the CCD camera (120 Hz). For comparison, we show data for 10.9 and 1.9
kbp DNA (upper and lower triangles, respectively) from ref. 42.
15412

www.pnas.org/cgi/doi/10.1073/pnas.1407197111 Lipfert et al.
multiple digoxigenin labels at the other end that enable attachment to strep
tavidincoated magnetic beads and the antidigoxigenin–coated surface, re
spectively (Fig. 1E). For control measurements on dsDNA, we used several
different constructs. Unless otherwise noted, we used 3.4 or 20.6kbp dsDNA
molecules that were ligated at their respective ends to ∼0.6kbp PCRgenerated
DNA “handles”that include multiple biotin or digoxigenin labels. To test
whether in particular the surprising differences in twist–stretch coupling and
buckling dynamics between dsRNA and dsDNA might be influenced by the fact
that our dsRNA construct carried labels on only one strand at each end whereas
the standard dsDNA constructs for MTs measurements carried labels on both
strands on both ends, we generated an alternative DNA construct with labels
on only one strand at each end (SI Appendix,Fig.S7A). The alternatively labeled
dsDNA construct behaved identically, within experimental error, to the con
ventional dsDNA constructs (SI Appendix,Fig.S7Band C), suggesting that the
labeling procedure does not affect the observed mechanical properties.
Measurements were conducted using custombuilt magnetic tweezers in
TrisEDTA (TE) buffer (Sigma; pH 8.0) containing 10 mM Tris·HCl and 1 mM
EDTA supplemented with SUPERase·In RNase inhibitor (Ambion; 0.1 U/μL
final concentration) and with varying amounts of NaCl added.
ACKNOWLEDGMENTS. We thank Bronwen Cross, Theo van Laar, and
Susanne Hage for technical assistance; Zhuangxiong Huang for help
with initial data analysis; and Aleksei Aksimentiev and members of the
Department of Bionanoscience for useful discussions. We acknowledge
funding from a Howard Hughes Medical Institute International Student
Research Fellowship, a Stanford BioX Graduate Student Fellowship,
a BurroughsWellcome Career Award at the Scientific Interface, Na
tional Institutes of Health Grant R01GM100953, the Delft University
of Technology, a VENI grant of the Netherlands Organisation for Scien
tific Research, the European Research Council, and a European Young
Investigator grant from the European Science Foundation.
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Lipfert et al. PNAS

October 28, 2014

vol. 111

no. 43

15413
BIOPHYSICS AND
COMPUTATIONAL BIOLOGY
1
Supplementary Information for:
DoubleStranded RNA under Force and Torque: Similarities
to and Striking Differences from DoubleStranded DNA
Jan Lipferta,b, Gary M. Skinnera,1, Johannes M. Keegstraa,2, Toivo Hensgensa, Tessa
Jagera, David Dulina, Mariana Köbera, Zhongbo Yua, Serge P. Donkersa, Fang
Chieh Chouc, Rhiju Dasc,d, and Nynke H. Dekkera,3
aDepartment of Bionanoscience, Kavli Institute of Nanoscience, Delft University of
Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
bDepartment of Physics, Nanosystems Initiative Munich, and Center for NanoScience
(CeNS), LudwigMaximiliansUniversity Munich, Amalienstrasse 54, 80799 Munich,
Germany
cDepartments of Biochemistry and dPhysics, Stanford University, Stanford, CA 94305,
USA
1Present address:
Illumina UK, Chesterford Research Park, Little Chesterford, Essex, CB10 1XL, United
Kingdom
2Present address:
FOM Institute for Atomic and Molecular Physics (AMOLF), Science Park 104, 1098 XG
Amsterdam, the Netherlands
3Corresponding author: Nynke H. Dekker
Email: N.H.Dekker@tudelft.nl
Phone: +31152783219
Supplementary Information
Tables S1S7
Figures S1S7
2
SUPPLEMENTARY MATERIALS AND METHODS
Doublestranded RNA constructs for magnetic tweezers experiments
Overview of the protocol. Here we give an overview of the protocol to generate the
dsRNA magnetic tweezers construct. In brief, our protocol to generate fully double
stranded RNA constructs is based on annealing two complementary singlestranded RNA
molecules that carry multiple biotin or digoxigenin labels at their respective 5’ ends (Fig.
1c). The product of the annealing reaction is a 4218 bp fully doublestranded RNA
construct with multiple biotin labels at one end and multiple digoxigenin labels at the
other end that enable attachment to streptavidincoated magnetic beads and the antidig
coated flow surface, respectively (Fig. 1e). Prior to the final annealing step, the two
singlestranded RNA molecules are prepared in separate reactions by T7 in vitro
transcription from DNA templates (Fig. 1c,d). The T7 transcription proceeds in two
steps. In an initial step, short sequences that contain only three of the four nucleotides are
transcribed in the presence of biotin or digoxigeninlabeled nucleotides and the
polymerase is stalled at the fourth nucleotide that is omitted from the reaction mixture.
The stalled polymerase complexes are purified and transcription is reinitiated in the
presence of all four unlabeled nucleotides to complete the singlestranded RNA
molecules (Fig. 1c).
Construction of the DNA templates. The DNA templates for RNA transcription are
generated by PCR from a pBAD plasmid using hotstart Herculase (Agilent) with
nucleotide concentrations of 200 nM (Promega) and appropriately chosen primers (see
below). The final DNA template for the biotinlabeled (digoxigeninlabeled) strands
consists of a T7 promoter followed by a 33 nt sequence that contains 12 A, but no T (12
T, but no A), in turn followed by a 4.2 kb sequence starting in T (starting in A). The
resulting PCR products are column purified using the Nucleospin Gel and PCR
purification kit (Macherey Nagel) according to the vendor’s protocol, and imaged by gel
electrophoresis.
Forward primer biotinlabeled strand:
5’TAATACGACTCACTATAGGGAGACCAGGACCAGACCAGGACCAGACCAG
GACCTAAGATTAGCGGATCCTACCTGAC 3’
Reverse primer biotinlabeled strand:
5’GGGTGTCCTGGTCCTGTCCTGGTCCTGTCCTGGTCCAGGTTAACCTCAA
CTTCCATTTCC 3’
Forward primer digoxigeninlabeled strand:
5’TAATACGACTCACTATAGGGTGTCCTGGTCCTGTCCTGGTCCTGTCCTGG
TCCAGGTTAACCTCAACTTCCATTTCC 3’
Reverse primer digoxygeninlabeled strand:
5’GGGAGACCAGGACCAGACCAGGACCAGACCAGGACCTAAGATTAGCGG
ATCCTACCTGAC 3’
Twostep T7 in vitro transcription. The purified PCR products are used as templates in
T7 in vitro transcription reactions. Initial transcription reactions are carried out using the
3
Ribomax Large Scale RNA production kitT7 (Promega) with a reaction mix containing
16 nM GTP, CTP, and biotinylatedATP (Perkin Elmer), but no UTP (16 nM GTP, CTP,
digoxigenatedUTP, but no ATP) and a 10fold reduced T7 polymerase concentration
(compared to the vendor’s protocol) for 10 min at room temperature. During the initial
transcription reactions, 12 biotin labels (12 digoxigenin labels) are incorporated and the
polymerase subsequently stalls at the missing fourth nucleotide. The stalled polymerase
complexes are purified twice on Illustra Microspin G25 size exclusion columns according
to the vendor’s protocol. To block free polymerases, we subsequently add 25 µg/µl
heparin (heparin sodium salt from porcine intestinal mucosa; Sigma) to the reactions and
incubate for 5 min at room temperature. To complete the 4.2 kb singlestranded RNA
molecules, we reinitiate the transcription reactions in the presence of 1.5 mM of all four
unlabeled NTPs in the reaction mixture and incubate for 1 h at room temperature. After
completion of the transcription reactions, we add DNase (1 unit/µg of template DNA),
incubating for 20 min at 37 ºC to fully digest the DNA templates. Subsequently, we
purify the singlestranded RNA molecules on RNeasy columns according to vendor’s
protocol (Qiagen). The resulting RNA concentrations are determined using a Nanodrop
photospectrometer (Isogen Life Sciences). Once the singlestranded RNA molecules are
complete, we anneal them in equimolar amounts (typically between 5001000 ng) in a
buffer containing 75 mM NaCl and 7.5 mM sodium citrate, with a final volume of 100 µl.
Hybridization is performed in an Eppendorf thermocycler (Mastercycler Personal) by
incubating for 60 min at 65 °C, and subsequently cooling down in steps of 1.2 °C and 1.3
°C alternating every 5 min to a final temperature of 25 °C to yield to the final double
stranded product. The resulting doublestranded RNA product is purified on a RNeasy
column, eluted in 1xTE buffer containing 1% ethanol, and stored at 80 ºC prior to use in
the magnetic tweezers.
Magnetic tweezers for singlemolecule measurements
Buffers for magnetic tweezers measurements. Measurements were performed in TE
buffer (Sigma), pH 8.0, containing 10 mM TrisHCl and 1 mM EDTA
(ethylenediaminetetraacetic acid), supplemented with SUPERase·In RNase inhibitor
(Ambion) at a final concentration of 0.1 unit/µl and with varying amounts of NaCl added,
unless otherwise noted.
Doublestranded DNA constructs with PCRgenerated “handles” for magnetic tweezers
measurements. For reference measurements on dsDNA in the MT, we employed either
3.4 kbp or 20.6 kbp dsDNA constructs with multiple biotin and digoxigenin labels at their
respective ends. Biotin or digoxigenin labeled dsDNA was generated in PCR reactions
that included labeled nucleotides and the labeled PCR products were attached to the
central, unlabeled dsDNA molecule by ligation, as described previously (1). The 20.6 kbp
dsDNA molecule (45.6% GC content) is based on the Supercos1lambda1,2 plasmid and
was used previously (2, 3); details of the protocol are given in Ref. (1). The 3.4 kbp
dsDNA (45.8% GC content) was selected to match the contour length of dsRNA and is
based on the pRLSV40 plasmid (Promega) digested with BamHI and XbaI and again
ligated to PCRgenerated DNA handles containing biotin or digoxigenin, respectively.
4
Doublestranded DNA constructs with labels on only one strand at each end. For control
measurements, we created a dsDNA construct that has biotin and digoxigenin labels for
bead and surface attachment, respectively, on only one strand at each end, similar to our
dsRNA construct and unlike the PCRgenerated standard dsDNA constructs for MT
measurements (described in the previous section). Singlestrand labelling of dsDNA was
achieved by singlestrand nicking followed by a Klenow fillin reaction. We started with
the pRLSV40 plasmid DNA (Promega Corporation, Madison, WI). To introduce
restriction sites for the nicking enzymes NtBbvCI and NbBsmI and the restriction
enzyme SmaI, forward oligo 5’P
GATCCCTCAGCGGGAGACCAGGACCAGACCAGGACCAGACCAGGACCCGGG
ACCAGGACAGGACCAGGACAGGACCAGGACACCCGAATGCG was annealed to
reversed oligo 5’P
CTAGCGCATTCGGGTGTCCTGGTCCTGTCCTGGTCCTGTCCTGGTCCCGGGTC
CTGGTCTGGTCCTGGTCTGGTCCTGGTCTCCCGCTGAGG and ligated into
BamHI and XbaIdigested pRLSV40. The resulting plasmid pRLSV40BbvCIBsmI
was amplified and subsequently linearized with SmaI and the top strand nicked with Nt
BbvCI (New England Biolabs, Ipswich, MA). The 5’ 42bpfragment was melted out and
filled in with Klenow using a mixture of nucleotides containing dTTP, dCTP, dGTP
(purchased from Promega Corporation, Madison, WI) and Bio14dATP (Invitrogen/Life
Technologies). After purification, the bottom strand of pRLSV40BbvCIBsmI was
nicked with the nicking enzyme NbBsmI (New England Biolabs, Ipswich, MA). The
42bpfragment was melted out and filledin with Klenow using a mixture of nucleotides
containing dATP, dCTP, dGTP (purchased from Promega Corporation, Madison, WI)
and Dig11dUTP (Roche Applied Science). Remaining nicks were closed by T4 DNA
ligase.
Magnetic tweezers set up. Our MT implementation has been described previously (1, 2,
4, 5). Briefly, a 100× oilimmersion objective (Olympus ACH 100X; numerical aperture
(NA) = 1.25) connected to a CCD camera (Pulnix TM6710CL) was used to image
superparamagnetic beads tethered by dsRNA molecules to the surface of a flow cell.
Flow cells were made from glass microscope cover slips with a double layer of parafilm
as a spacer. The bottom surface was coated with nitrocellulose (0.1% (wt/vol) in amyl
acetate) and flow cells were stored dry. Before measurements, flow cells were
extensively rinsed with RNaseZap (Invitrogen), followed by rinsing with milliQ water,
and rinsing with TE + 200 mM NaCl buffer. 3.0µmdiameter nonmagnetic latex beads
(Invitrogen) were aspecifically attached to the bottom surface by incubation in TE + 200
mM NaCl buffer for 30 min to act as reference beads. Before addition of the RNA
construct to the flow cell, the bottom surface was functionalized by incubation with 100
µg·ml−1 antidigoxigenin (Roche) in PBS buffer (Sigma) for 60 min to provide for RNA
attachment and was passivated by incubating for 30 min with 2 mg·ml−1 bovine serum
albumin (Sigma) in TE + 200 mM NaCl buffer. The functionalized RNA constructs were
incubated in the flow cell for 30 min at a final concentration of ~0.1 ng/µl in TE + 200
mM NaCl buffer. Streptavidincoated superparamagnetic MyOne beads (Invitrogen) or
M270 beads (Invitrogen) were diluted 50fold in TE + 200 mM NaCl buffer, flushed into
the flow cell, and incubated for 30 min. Finally, unattached beads were flushed out with
TE + 200 mM NaCl buffer.
5
The positions of a dsRNAtethered bead and a reference bead attached to the surface
were tracked simultaneously at a rate of 120 Hz. From analysis of the CCD images, the
bead positions in x, y and z were determined (6, 7). After subtraction of the reference
bead position to correct for mechanical drift, the tethered bead was tracked with an
accuracy of ~12 nm in the x, y and z dimensions.
Force calibration in the magnetic tweezers. We determined the stretching force applied
in the MT (i.e. the magnetic force pulling the bead away from the surface) from analysis
of the bead’s fluctuations, using the relationship (8):
F = L·kBT / Var(x) (1)
where L is the tether extension, determined as the mean of the zposition above the
surface, kB is Boltzmann’s constant, T the absolute temperature and Var(x) is the variance
of the fluctuations in the x (i.e. inplane) position. In order to determine Var(x)
accurately, biases due to the finite acquisition speed of the CCD camera need to be taken
into account (7, 911). We utilized the method that determines the force from the
integrated power spectral density of the xfluctuations using iteratively applied
corrections for the finite camera acquisition frequency (10). Control calculations, using a
method that analyzes the power spectral density using a closedform expression to
account for corrections and a method that analyzes the fluctuations in real space using the
Allan variance (11) gave identical results, within experimental error (Fig. S1a). For the
rotationextension measurements (Figs. 2, 4, and 5) and for dynamic force spectroscopy
measurements (12) (Fig. S2c,d), we used the forces from predetermined relationships of
magnetic position and applied stretching force for our experimental configuration (Fig.
S1b,c).
Magnetic tweezers for torque measurements. The magnetic torque tweezers (MTT) are a
variant of the magnetic tweezers in which the pair of rectangular magnets (Fig. 1d) is
replaced by cylindrical permanent magnet to apply forces and a smaller side magnet to
apply torques (13) (Fig. 3a and Fig. S4a). Alternatively, two pairs of Helmholtz coils
arranged in (x,y)plane can be used to apply torques, an approach termed electromagnetic
torque tweezers (eMTT) that allows one to set the torsional stiffness of the angular trap
independently of the magnitude of the applied stretching force (14) (Fig. S4b). The
torque measurement relies on tracking the rotation angle
θ
of the bead about the zaxis,
i.e. the nucleic acid tether axis (13). The torsional trap stiffness k
θ
was calibrated for each
measurement from the variance of the rotational fluctuations Var(
θ
):
k
θ
= kBT / Var(
θ
) (2)
If a torsionally constrained nucleic acid tether is over or underwound away from its
torsionally relaxed equilibrium angular position (
θ
0), the resulting restoring torque leads
to a shift in the mean of the angular fluctuations
Δθ
=
〈θ
N 
θ
0
〉
, where
θ
N is the angle
position after N turns and 〈…〉 denotes the mean (13). The restoring torque exerted by the
nucleic acid tether was calculated as:
6
Γ
=  k
θ
·
Δθ
(3)
The rotation angle can be monitored either directly through the use of an angular marker
and image analysis (13, 15) or indirectly by converting the (x,y)position of the bead to
angular and radial coordinates (14, 16) (Fig. S4c,d). Torque measurements were carried
out both using the MTT set up described by Lipfert et al. (13) and the eMTT instrument
described by Janssen et al. (14). In both cases, we employed the angle tracking protocol
based on conversion from (x,y)position and custommade cylindrical magnets consisting
of a stack of 6 magnets each 1 mm in height (for 6 mm total height), 6 mm in diameter
and with a central aperture of 1 mm in diameter. For some measurements, a cylindrical
magnet was used in which the last magnet in the stack was assembled with opposite
magnetization direction; this “flipped” magnet assembly has been shown to give larger
forces than a similar magnet stack where all magnets have the same magnetization
direction (17). Forces in the MTT and eMTT were calibrated as described in the “Force
calibration in the magnetic tweezers” section, except that the variance of the radial
component of the fluctuations was used instead of the xposition (16).
Elastic rod model for dsDNA and dsDNA
Isotropic rod model of polymer elasticity. The elasticity of twiststoring biopolymers can
be modeled in the framework of the isotropic rod model (18). The isotropic rod model
has been, in particular, used as a coarse grained model for dsDNA that neglects specific
sequences effects and is expected to be valid on length scales much longer than one base
pair. The deformations of a segment of an isotropic rod (Fig. 1b) can be described by
three quantities: 1) the stretch or extensional deformation u(s) that measures the fractional
change in the length of the segment, 2) the bend or bending deformation
β
(s) that
measures how the tangent vector t(s) changes along the rod, and 3) the twist density or
torsional deformation
ω
(s) that determines how the each segment is rotated around the
axis of the rod with respect to adjacent segments, where s is denotes the coordinate along
the rod. The total elastic energy of the rod is given by integrating contributions dE(s)
along its total length (18):
E = ∫ dE(s)ds = kBT/2 ∫0Ltot (Aβ2 + Bu2 + Cω2 + 2Duω)ds (4)
where the respective terms in the rightmost integral in turn represent contributions from
bending energy, stretching energy, twisting energy, and twiststretch coupling energy.
Note that the upper limit of the integrand equals Ltot, the total length of the stretched rod,
which may exceed the contour length LC. Each term comes with a phenomenological
coupling constant: A is the bending persistence length (in units of length), B is the stretch
modulus (in units of inverse length), which is more commonly expressed as S = B·kBT
(where S is the stretch stiffness in units of force), C is the torsional persistence length (in
units of length), and D is the (dimensionless) twiststretch coupling.
Inextensible and extensible WLC models. A further simplification of Eq. 4 is the
inextensible wormlike chain (WLC) model that assumes the polymer to be torsionally
unconstrained and inextensible (1921). The elastic energy in the inextensible WLC
7
model simplifies to the first term, i.e. the bending energy term, in the integral in Eq. 4.
The inextensible WLC model provides an accurate description of the stretching behavior
of dsDNA (1921) and dsRNA (22, 23) in the absence of torsional strain and for low
forces, i.e. in the limit that F « B·kBT, the socalled enthalpic stretching regime. The WLC
model has been solved numerical to yield the force F as a function of the molecule’s
extension z and a number of approximation formulae exist. In this work, we use the
seventh order approximation to the numerical solution due to Bouchiat et al. (24):
F(z) = kBT/A · [1/4(1−z/LC)2 – ¼ + z/LC + Σi=2,…,7 αi (z/LC)i ] (5)
in which the contour Lc and the bending persistence length A are treated as free
parameters. The
α
i are numerical coefficients given in Ref. (24). For higher forces, F > 5
10 pN for dsDNA and dsRNA, elastic stretching contributions become relevant and the
forceextension data can be described using the extensible WLC model (2427). In the
extensible WLC model, the terms z/LC in Eq. 5 is replaced by z/LC – F/S (Ref. (24)).
Models of dsDNA and dsRNA under torsional constraint. Fully doublestranded DNA or
RNA molecules that are free of nicks and attached via multiple attachment points at both
ends can experience torsional strains, giving rise to a complex forcetorque response. A
useful quantity to describe twiststoring polymers under torsional constraint (6, 28) is the
linking number Lk. The linking number is a topological invariant for torsionally
constrained molecules (2931) and partitions into twist Tw and writhe Wr:
Lk = Tw + Wr (6)
Essentially, Tw is the number of turns in the double helix and Wr is the number of times
the helical axis crosses itself. Magnetic tweezers and magnetic torque tweezers control Lk
of the molecule under study. It is convenient to consider the linking number with respect
to the torsionally relaxed molecule: this is the definition for the number of applied turns
used throughout the text (Figs. 25 and Fig. S3), i.e. zero turns corresponds to a
torsionally relaxed molecule. For a torsionally relaxed molecules Lk0 = Wr0 + Tw0 with
Wr0 = 0 and Tw0 being equal to the natural twist of the double helix, i.e. the number of
base pairs divided by ~10.5 base per turn for dsDNA and ~11.3 base pairs per turn for
dsRNA (32). Another useful quantity in this context is the supercoiling density, defined
as
σ
= (Lk  Lk0)/Lk0, which is normalized to the natural twist of the molecule (Figs. 24
and Fig. S3k,l).
For small deviations of the linking number away from the torsionally relaxed equilibrium
state, the change in linking number is initially absorbed by elastic twist deformations for
both dsDNA (8, 13, 15, 33, 34) and dsRNA (Fig. 3). In this regime, the torque increases
linearly with the number of applied turns N:
Γ
= 2π·N·kBT·C / LC (7)
While it is possible to include nonlinear terms into the twist response close to zero
applied turns, the current data are welldescribed by a linear model (13, 33, 35) (Fig. 3c).
8
C in Eqn. 7 is the effective twist persistence length, since bending fluctuations decrease
the effective twist persistence compared to its intrinsic value (36, 37). Moroz and Nelson
have developed a model of the forcedependence of the effective torsional stiffness (36,
37). They use a perturbative approach, valid in the highforce limit; to third order in F1/2,
their model gives (16, 38):
C=Clim 1−Clim /A
4AF
kBT
"
#
$%
&
'
1/2 +Clim /A
( )
2−2Clim /A
( )
16 AF
kBT
"
#
$%
&
'
−4Clim /A
( )
3−16 Clim /A
( )
2+21 Clim /A
( )
256 AF
kBT
"
#
$%
&
'
3/2
"
#
$
$
$
$
$
%
&
'
'
'
'
'
(8)
where Clim is the intrinsic twist persistence length, adopted in the highforce limit. While
Eqn. 8 provides a reasonable qualitative description of the data, we observe deviations
from the model at low forces (< 2 pN) for both dsDNA (13, 16) and dsRNA (Fig. 3f),
likely, at least in part, due to the highforce perturbative approach of the model. We
obtain Clim from fits of Eqn. 8 to the C(F) data at forces > 2.5 pN; this cutoff is an
empirical choice, giving values for Clim of dsDNA consistent with previous extrapolations
to high stretching forces as well as with direct measurements at high forces (13, 33).
DsRNA or dsDNA molecules that are twisted further away from their torsionally relaxed
equilibrium conformation beyond the linear torque response regime, undergo
conformational changes, giving rise to a complex forcetorque phase diagram (Fig. 3g).
While a number of models have been proposed to describe aspects of this forcetorque
diagram for dsDNA (see e.g. (3949)), there is currently no commonly accepted model
that quantitatively accounts for all aspects of the diagram. Here, we limit the discussion
to relatively simple models that account for the formation of plectonemic supercoils as
the linking number is increased beyond the critical supercoiling density for buckling.
A basic model of plectonemic supercoiling considers the twist energy and the energy
required to form a circular loop (6, 28). This simple model makes predictions for the
buckling torque
Γ
buck, the slope of the extension vs. turns response in the plectonemic
regime
Δ
L/turn, and for the number of turns at which buckling occurs Nbuck:
Γ
buck = (2A·kBT·F)1/2 (9)
Δ
L/turn =
π
· (2A·kBT / F)1/2 (10)
Nbuck ≈ LC·(A·F/ (2
π
2·C2·kBT))1/2 (11)
We note that these model predictions do not have any free parameters, since A, LC, and C
can be determined independently from forceextension measurements (Fig. S2 and Fig.
S1df), and torqueturn measurements (Fig. 3c), respectively. The simple model gives the
right trends but only qualitative agreement with our measurements for
Γ
buck,
Δ
L/turn, and
Nbuck (dashed lines in Fig. 2b,c and Fig. 3e). Quantitative deviations are to be expected,
9
given the many approximations inherent in the model, including the neglect of thermal
fluctuations and the consideration of a circular loop only.
A more refined model is due to Marko (50), which includes an additional parameter, the
torsional stiffness of the plectonemic state P. The expressions for the buckling torque
Γ
buck, the slope of the extension vs. turns response in the plectonemic regime
Δ
L/turn, and
for the critical supercoiling density
σ
buck in the Marko model (following the notation of
Wang and coworkers (34)) are:
Γ
buck = (2·kBT·P·g / (1P/C))1/2 (12)
Δ
L/turn =
2
π
ω
0
1−1
2
kB
T
Af −
ω
0
2C2
16
kB
T
Af
"
#
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&
'
3/2
1
c
2pg
1−p/c
"
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&
'
2
(
)
*
*
+
,


2pg
1−p/c
1
p−1
c
"
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&
'
(13)
σ
buck = 1/c · (2·p·g / (1P/C))1/2 (14)
with g = F – (F·kBT / A)1/2, C is the torsional twist stiffness, approximated by the model
of Moroz and Nelson, and p = kBT·P·
ω
02 and c = kBT·C·
ω
02 are scaled quantities related
to P and C (where
ω
0 is 2
π
divided by the helical pitch). We fit the value of P
independently to the critical supercoiling density data (Fig. 2b), to the slopes in the
plectonemic regime (Fig. 2c), and to the critical torque data for buckling (Fig. 3e) for
dsRNA and obtain consistent results, within experimental error, with PRNA ~ 22 nm. We
use the model by Brutzer et al. (51) that essentially extends the treatment due to Marko
(50) to analyze the buckling transition (see the “Twostate model of the buckling
transition” section).
Determination of the twiststretch coupling from the slope of rotationextension curves.
The twiststretch coupling parameter D can be determined from the change in extension
upon over and underwinding the molecule (5254). At forces sufficiently large to
suppress strong bending fluctuations, the energy per length of a stretched and twisted
dsRNA or dsDNA molecule can be expressed by modifying Eqn. 4 and adding a term –
F·u to account for the energy contribution of the external force:
E/L = kBT/2 · (Aβ2 + Bu2 + Cω2 + 2Duω) − F·u (15)
Minimize the energy with respect to u at constant ω and F, we find
kBT/2 · (2Bu + 2Dω) – F = 0 (16)
Considering now two measurements at the same force, one at an arbitrary value of the
twist
ω
and a second for a torsionally relaxed molecule
ω
= 0, we subtract the
expressions and obtain:
10
B(u – uω=0) + Dω = 0 (17)
The slope of the rotation extension curve is then given by
d(u – uω=0)/dω = −D/B (18)
Alternatively, expressing the slope as the change in length ΔL per applied turns, we can
write
dΔL/dN = −2πD/B (19)
This then gives an expression for D equal to:
D = −(dΔL/dN)·B / (2π) = −( dΔL/dN)·S / (2π·kBT) (20)
The uncertainty Δ(D) in D is given by standard propagation of errors:
Δ(D)/D = ( [Δ(dΔL/dN) / (dΔL/dN)]2 + [Δ(S)/S ]2 )1/2 (21)
We note that instead of the dimensionless twiststretch coupling parameter D that we use
in this work, some authors employ g = kBT·D (with units of energy; see, e.g., Ref. (53))
or D’ = D/(1.85/nm) (with units of length; see, e.g., Ref. (52, 54)). Here, we consistently
employ the dimensionless convention for D.
Models for the twiststretch coupling and experimental results for dsDNA. Naïve intuition
suggests that a helical rod shortens as it overwound, or, equivalently, unwinds as it is
stretched. Early work based on interpretation of singlemolecule stretching and twisting
experiments suggested that dsDNA indeed shortens upon overwinding, i.e. that it exhibits
a positive twiststretch coupling D (Refs. (52, 55)). Marko (55) analyzed the DNA
overstretching data of Cluzel et al. (56) in the framework of an elastic model with a
stable overstretched state and found a value of DDNA = 30. Kamien et al. (52) analyzed
the DNA length response upon overwinding in the magnetic tweezers data of Strick et al.
(8) and determined DDNA = 22, similarly with a positive sign. The apparent positive twist
stretch coupling was explained in terms of simple elastic models that described dsDNA
as an isotropic material with a helical shape (52, 55). Similarly, a higher resolution model
with coarse–grained representations for each base found a positive D for dsDNA (57). In
addition, atomistic molecular dynamics simulations also found positive values for the
twiststretch coupling, in the range of DDNA = 7.5 – 33 (Ref. (58)).
However, the initial estimates of D from experimental data included regions where the
dsDNA underwent structural transitions, such as melting and overstretching, and were
not representative of the behavior of dsDNA close to its unperturbed Bform structure.
Higher resolution MT experiments found that dsDNA lengthens upon overwinding,
indicative of a negative D value (53, 54). Lionnet et al. (54) found a slope of 0.42 ± 0.2
nm/turn, independent of salt concentration and stretching force up to ~15 pN, and
deduced DDNA = 16 ± 7. Gore et al. (53) obtained a similar value for the length increase
upon overwinding of 0.5 ± 0.1 nm/turn, corresponding to DDNA = 22 ± 5. In addition,
11
Gore et al. employed a rotor bead assay to directly monitor the change in equilibrium
twist upon stretching of DNA and found that stretching by 1% leads to an increase in
twist by ~0.1%, in agreement with the value for the twiststretch coupling obtained from
the magnetic tweezers measurements (53). Using the rotor bead assay, they found that for
large stretching forces (> 30 pN) the twiststretch coupling changes sign and becomes
positive. Recently, Sheinin and Wang (59) employed an optical torque wrench assay to
simultaneously monitor the extension and torque upon overwinding DNA. They analyzed
their data in the framework of a model that goes beyond the linear extension response to
winding and includes the effect of bending fluctuations, allowing a fit of the data over a
larger range of σ values. They found DDNA = 21 ± 1, independent of force up to ~10 pN
(Ref. (59)), consistent with the earlier measurements by Gore et al. and Lionnet et al. In
this work, we limited our analysis of the rotation response to the range 0.01 < σ < 0.02
and found that approximating the DNA response as linear only leads to small corrections
compared to the full nonlinear model.
A negative twiststretch coupling can be rationalized by models that include a stiff helical
backbone wrapped around a softer core material (53, 60). In addition, several higher
resolution models with atomistic representation of dsDNA were consistent with negative
values for the twiststretch coupling (54, 61, 62).
Models for the twiststretch coupling of RNA. Experimentally, we found a positive twist
stretch coupling for dsRNA, based on the observation that dsRNA shortens by 0.85 ±
0.04 nm per turn upon overwinding. We note that including the nonlinear terms as
described by Wang and coworkers (59) did not significantly affect our results. How can
this surprising result be understood in terms of molecular models? The observed positive
twiststretch coupling for dsRNA could be rationalized in terms of the simple elastic
models of a helical isotropic material initially proposed for dsDNA (52, 55). In essence, a
helix with a constant radius must shorten as it is overwound (53, 54). However, this
approach is unsatisfying since it evidently fails for dsDNA.
The experimental observation of a negative twiststretch coupling for dsDNA led to the
proposal of models that feature a stiff backbone wrapped around an isotropic core (53,
60) and involve a change in helix radius upon overwinding. These stiff backbone models
correctly account for the negative twiststretch coupling for dsDNA. However, they
predict the twiststretch coupling of dsRNA to also be negative. For example, the “toy
model” by Gore et al. (53) predicts an even more pronounced negative twiststretch
coupling for dsRNA compared to dsDNA due to its smaller helix angle, in clear contrast
to our experimental results.
The apparent failure of simple elastic models to satisfactorily account for the twiststretch
coupling of both dsDNA and dsRNA in a unified framework might suggest that at least
some of the microscopic details of the molecules need to be taken into account for
quantitative predictions of D. Kosikov et al. simulated dsDNA polyAT and polyGC
homopolymers, employing allatom potentials for the nucleic acid and an implicit
treatment of the solvent and ion atmosphere in a molecular mechanics framework (63).
These authors found, for conformations close to the equilibrium values for the helical
rise, that Aform dsDNA untwists upon stretching and Bform dsDNA overtwist upon
stretching, in qualitative agreement with our results for dsRNA and dsDNA.
Quantitatively, their constant stretch, variable twist simulations predict slopes of ~ −3º
12
per 0.1 nm stretching for Aform dsDNA and ~ +3º per 0.1 nm stretching for Bform
dsDNA (where the exact values depend on the fitting range and conformational family
considered). In order to determine the value of the twiststretch coupling D from these
slopes that were obtained in the imposed stretch, variable twist ensemble, one needs to
consider an argument similar to the derivation outlined in Eqns. 1520. However, in
contrast to the imposed twist, variable stretch ensemble considered in Eqns. 1520, the
minimization is with respect to the twist ω, at constant u. The result analogous to Eqn. 18
in the imposed stretch, variable twist ensemble is D = −C·(Δω/Δu). Consequently, the
prediction from the slopes are DDNA~ −50 and DRNA~ +50. These predictions have the
correct sign but are in absolute value too large by a factor of 35 in magnitude, compared
to the experimental results.
We have modeled dsRNA and dsDNA using the framework of six basestep parameters
(see the “Basestep parameter model” section), which presents an intermediate resolution
model. Ultimately, a full understanding of the striking differences in the twiststretch
behavior might require modeling with a full atomistic descriptions of the nucleic acids,
ions, and solvent and presents an interesting challenge for molecular dynamics
simulations or related approaches.
Twostate model of the buckling transition. We use the twostate model by Brutzer et al.
(51) to analysis the tether extension data at the buckling transition and to determine the
characteristic timescale for buckling
τ
buck (Fig. 5). Briefly, prior to buckling after N
applied turns the free energy of the DNA is given by
Epre(N) = 0.5·C/LC ·(2π)2·N2 (22)
The free energy after buckling is equal to
Epost(N) = Eb + 0.5·C/LC ·(2π)2·(N − ΔNb)2 (23)
where Eb is the energetic penalty that must be overcome for the formation of the buckling
structure and
Δ
Nb is the amount of twist (in number of turns) that is transferred into
writhe during buckling. The probability that the postbuckling state is occupied ppost is
given in the framework of this simple twostate model by Boltzmann statistics:
ppost = 1/(1+exp[(Epost − Epre)/kBT]) (24)
Inserting the expression for Epre and Epost the probability can be written as:
ppost = 1/(1+exp[(C/LC ·(2π)2·(Nb − N)·ΔNb)/kBT]) (25)
where Nb equals the number of applied turns at the point of buckling equilibrium, i.e. the
number of turns where Epre = Epost. By fitting the dependence of the postbuckling
population on the number of applied turns to this expression for the probability, we
determined both Nb and
Δ
Nb (Fig. 5c and Fig. S5l). Notably, we find that the fitted values
for
Δ
Nb for our 4.2 kbp dsRNA tethers are consistently larger than one, i.e. that more than
one full turn is converted from twist to writhe at the buckling transition, similar to what is
13
observed for dsDNA (Ref. (51) and Fig. S5l). In addition, we find
Δ
Nb for dsRNA to
increase weakly with increasing force and salt concentration, again qualitatively similar
to what has been observed for dsDNA (51).
To describe the dependence of the mean residence times in the pre and postbuckling
states,
τ
pre and
τ
post, on the number of applied turns in the twostate framework, we
assume an Arrhenius relationship with an exponential dependence on the number of
applied turns (51). The expression for the mean residence time of the prebuckling state is
τpre = τbuck · exp[−(C/LC ·(2π)2·(Nb − N)·ΔNpre)/kBT] (26)
where
τ
buck is the overall characteristic residence time at the buckling transition and
Δ
Npre
is the angular distance to the transition state from the prebuckling state. A similar
expression holds for
τ
post, only with
Δ
Npre replaced by –
Δ
Npost, the angular distance to the
transition state from the postbuckling state. Fits of the simple exponential, Arrhenius
like dependence to the pre and postbuckling residence times (Fig. 5c) were used to
determine the overall characteristic residence time
τ
buck as a function of applied force and
salt concentration (Fig. 5e).
The estimates of
τ
pre,
τ
post, and consequently
τ
buck are possibly biased due to the limited
time resolution of our instrument and due to the need to filter the data prior to applying a
threshold (Fig. S5ad). To estimate the effects of the finite sampling frequency, we
analyzed the data using sliding average filters of different width in the range of 40 to 10
Hz. For the dsRNA buckling data, using the filters of different width did not affect the
results for
τ
buck, within experimental error. In addition, we tested the effect of correcting
τ
pre and
τ
post for the finite acquisition time by applying a statistical correction method
based on the moment equations for a twostate Markov model (64). In brief, given
observed values
τ
pre and
τ
post measured with a detection limit of time
ξ
(which is set by
the camera frequency and width of the filter), the corrected “true” values
τ
*pre and
τ
*post
are determined by (numerically) solving the two equations (64):
τpre = (τ*pre + τ*post)·exp(ξ /τ*post) − τ*post (27)
τpost = (τ*pre + τ*post)·exp(ξ /τ*pre) − τ*pre (28)
While fitting the corrected values for
τ
pre and
τ
post (Eqn. 27 and 28) gave slightly lower
buckling times
τ
buck, compared to fitting the uncorrected data, the results were within
experimental error and differed at most by a factor of two for the fastest dsRNA buckling
times (Fig. S5h). In contrast, the dwell times in the pre and postbuckling states are
much smaller for dsDNA, compared to dsRNA (Fig. 5d). For the dsDNA data, changing
filter settings and applying the corrections for the finite detection limit both tended to
significantly affect the resulting values for
τ
buck. Therefore, we only report an upper limit
for the characteristic buckling for dsDNA at high force and high salt (F = 4 pN and 320
mM monovalent salt; dashed line in Fig. 5e). This upper limit of 50 ms is consistent with
14
the buckling times obtained previously for dsDNA molecules of different lengths (51)
(Fig. 5e, blue triangles)
Basestep parameter model
As a step beyond the simple isotropic elastic rod model (Eqn. 4), we have built a base
pair level model based on the six basestep parameters (65) slide, shift, rise, twist, roll,
and tilt (Fig. S6a, insets) for dsRNA and dsDNA. Basepair level models are intermediate
between elastic models that treat dsDNA or dsRNA as continuous rods and full atomistic
models. Following the approach of Olson and coworkers (66), we have determined the
average values and elastic couplings of the basestep parameters by analyzing their
observed values in highresolution crystal structures of nucleic acids deposited in the
protein data bank (http://www.rcsb.org/pdb/) using the program 3DNA (67). To ensure
data quality, we have only included structures with a resolution of better or equal to 2.8
Å. In addition, we eliminated strongly deformed structures by excluding parameter values
that were further than four standard deviations from the mean. To test the sensitivity of
our predictions on the data set used, we ran calculations using only structures that did not
contain bound proteins (“2.8_noprot” in Table S2) as well as control calculations using a
larger data set that included structures with bound proteins (“2.8_all” in Table S2).
Additional tests, calculations, and the numerical implementation of our basestep
parameter based modeling will be published elsewhere (F.C.C., J.L., and R.D., PLOS
Computational Biology, in press).
Comparison of dsRNA and dsDNA geometry. From the distribution of basestep
parameters, several observations can be made (Fig. S6a). First, the shift and tilt
distributions are similar for dsDNA and dsRNA, with average values near zero. Second,
dsRNA basesteps have, on average, a negative slide and positive roll, compared to
dsDNA basesteps that have approximately zero slide and roll. Third, both dsRNA and
dsDNA have positive values for rise and twist, with dsRNA taking on smaller values for
these two parameters, on average, compared to dsDNA.
These parameter differences at the level of basepair steps correspond to the geometric
differences observed between idealized Bform dsDNA and Aform dsRNA (see, e.g.,
Fig. 1a). In particular, the negative slide and positive roll values for dsRNA lead to the
axis that connects the base pair centers tracing out a pronounced helix, with a diameter of
~8 Å (Fig. S5b). In contrast, the center axis for dsDNA, which has on average close to
zero slide and roll, is approximately straight (with a helical diameter of only 0.6 Å; Fig.
S6b). This helical wrapping or “springiness” of the RNA centerline has several
interesting consequences. One implication of this structural difference between dsDNA
and dsRNA helices is related to the fact that the basestep parameter rise is not, in
general, the same as the “helix length per base pair” often also referred to as (helical)
“rise”. The average value for the basestep parameter rise is only slightly smaller for
dsRNA than for dsDNA (3.22 Å and 3.30 Å, respectively, for the “2.8_noprot” data sets;
Fig. S6a). In contrast, the helical rise is approximately 20% smaller for dsRNA compared
to dsDNA (2.8 Å for dsRNA (22, 23, 68, 69) and 3.33.4 Å for dsDNA; see e.g. Ref. (70)
and references therein). This difference can be understood from the “springiness” of the
RNA centerline; for helices with an (almost) straight centerline, such as dsDNA, the
helical rise and the basestep parameter rise are almost identical (and sometimes used
15
interchangeably). For a “springy” helix, such as dsRNA, the basestep rise does not
contribute fully to advancing the helix along its lengths and the helical rise is, therefore,
smaller than the basestep rise.
We note that there is a similar distinction between the basestep parameter twist and the
overall twist of the helix, i.e. the quantity Tw in Eqn. 6. This difference was pointed out
by Olson and coworkers (71) and taken into account in our calculations. Finally, a
comparable distinction has to be made between the twistrise covariance of the basestep
parameters and the overall twiststretch coupling of the helix (see below).
Simulations of singlemolecule experiments based on the basestep parameter model. We
carried out simulations based on a Metropolis Monte Carlo sampling scheme that mimic
singlemolecule stretching and twisting measurements in the MT. For dsDNA, our results
are in agreement with previous computational schemes for dsDNA (e.g., through normal
mode analysis (72)). Prior modeling of dsRNA is not available, and we chose to carry out
direct simulations to ensure rigorous calculation of experimental observables such as
global helix linking number, which cannot be computed simply as the sum of basepair
twist, as described above. The calculations are implemented in a software package
HelixMC that are being described in detail in a separate publication (Ref. (73); full
documentation publically available at http://github.com/fcchou/helixmc). This model
makes several simplifying assumptions: i) crystal structures present an accurate sampling
of flexibility in solution, ii) the distributions of the six nearestneighbor basestep
parameters give an adequate representation of the molecules, and iii) a purely harmonic
elastic treatment is sufficient to capture the fluctuations in solution. Given these
assumptions, only approximate agreement with experiments can be expected (74).
From simulated forceextension curves, we determined the bending persistence length
and stretch modulus of basestep level model dsDNA and dsRNA molecules (Table S2).
Similarly, we determined the torsional stiffness and twiststretch coupling from simulated
rotationtorque and rotationextension curves (Table S2). Comparing the results for the
“2.8_noprot” and “2.8_all” parameter sets gives a rough estimate of the robustness of our
simulation results. In general, the elastic stiffness parameters obtained from the
simulations tend to be lower for the “2.8_all” parameter set compared to the “2.8_noprot”
parameter set. A possible reason for the slightly lower observed stiffnesses when protein
bound structures are included is that the protein bound structures exhibit larger
deformations, compared to the proteinless structures, corresponding to larger local
flexibility.
The predictions for the bending persistence length are in close to quantitative agreement
with the experimental results for both dsDNA and dsRNA (Table S2). In addition,
simulations with both parameter sets correctly predict ARNA to be ~20% larger than ADNA.
The model predictions for the stretch modulus are a factor of ~2 larger than the
experimentally measured values for both dsDNA and dsRNA (Table S2). Nevertheless,
the model correctly predicts SRNA to be ~23 fold lower than SDNA. The difference
between the stretch modulus between dsDNA and dsRNA originates from the
“springiness” of dsRNA. When the helices are stretched, for dsDNA the applied force
mostly goes to the increase of rise; but for dsRNA the force can affect both rise and roll,
making dsRNA more pliable to global stretching than dsDNA (Table S3).
16
Similarly, while the absolute values of the predicted torsional stiffness C deviate from the
experimental results by a factor of ~2, the model correctly predicts them to be of similar
size for dsRNA and dsDNA, in particular when considering the “2.8_all” parameter set
(Table S2). Again, given the assumptions of the modeling, limitations in quantitative
agreement were expected.
Nevertheless, we were surprised that predictions of twiststretch coupling D were in
qualitative disagreement with the experiment. While the basestep model predicted the
slope in the twiststretch coupling regime for dsDNA within a factor of two (Table S2), it
predicts dsRNA to lengthen upon overwinding, in disagreement with our experimental
findings.
To produce better working models for dsDNA and dsRNA consistent with our available
data, we sought to understand the effect of each parameter in the covariance matrix of the
multivariate Gaussian distribution of the six basestep parameters. Due to nonlinearities
in relating these microscopic parameters to the elastic rod parameters, understanding
these effects required carrying out further simulations after doubling, halving, and
reversing the sign of each of the 21 independent parameters in the covariance matrix
(Tables S4 and S5). These calculations showed that the bending persistence length A is
predominantly affected by the variance of tilt and roll, as would be intuitively expected.
We also confirmed that the dominant factor controlling torsional persistence length C is
the variance of twist. Finally, the twiststretch coupling of dsDNA is affected by the
twistrise covariance. However, it appears that the twiststretch coupling D of dsRNA
receives substantial contributions form multiple factors, including the covariance of
twistrise, twistroll, twistslide, and the variance of twist and roll. This sensitivity of D to
multiple microscopic parameters is connected to the “springiness” (displaced helical axis)
of dsRNA, such that the effect of stretching is shared by changes in rise, slide and roll
and the twiststretch coupling is acutely sensitive to small changes in the twistrise, twist
roll and twistslide covariances. These covariances are themselves small numbers
(compared to intrinsic variances of twist, slide, and roll; Table S6). This analysis
underscores the importance of not conflating measurements of twiststretch coupling D
with the single basesteplevel covariance of twist and rise, which have distinct meanings
from the experimentally probed global ‘twist’ and ‘stretch’, respectively.
Based on the analysis of the effects of the parameters in the covariance matrix (Tables S4
and S5), we found that modest modifications of the covariance matrix led to accurate
recapitulation of all experimental measured mechanical properties (Tables S6 and S7). In
particular, the simulations now correctly recovered the positive twiststretch coupling of
dsRNA and gave detailed working models of how both nucleic acid helices responded to
force and torque (Fig. 4d). These working models are not uniquely defined; the greater
number of degrees of freedom in the base pair level covariance model (21) compared to
experimental observables (4) necessarily imply that other parameter sets could account
for the data as well. Future experimental efforts will be required to pinpoint correct
covariance parameters for dsDNA and dsRNA in solution. The current working basepair
level models, which reflect and recover available singlemolecule data (Tables S6 and
S7; Figure 4d), can be integrated efficiently via simulation to give quantitative
predictions for observables of other experimental methods such as NMR and Xray
scattering.
17
TABLES
Table S1: Elastic parameters of dsRNA and dsDNA from singlemolecule
measurements
Parameter
Symbol
(units)
dsRNA,
this work
dsRNA,
literature
dsDNA,
this work
dsDNA, literature
Bending
persistence length
A (nm)
57 ± 2
59.4 ± 2.9
Ref. (23)
61 ± 3
Ref. (23)
45 ± 2
47 ± 2 Ref. (75)
43 ± 3 Ref. (34)
44 ± 3 Ref. (13)
47.4 ± 4.4 Ref. (23)
49 ± 2 Ref. (23)
Stretch modulus
S (pN)
350 ± 100
500 ± 29
Ref. (23)
1000 ± 200
1087 ± 94 Ref. (76)
1401 ± 313 Ref. (27)
884 ± 116 Ref. (75)
1266 ± 217 Ref. (75)
935 ± 121 Ref. (23)
Torsional
persistence
length†
Clim (nm)
100 ± 2
109 ± 4
109 Ref. (36)
100 ± 7 Ref. (33)
107 ± 9.8 Ref. (33)
94 ± 7 Ref. (35)
109 ± 4 Ref. (13)
Slope of the
extension vs.
turns response
close to zero
turns
d
Δ
L/dN
(nm/turn)
0.85 ± 0.04
0.44 ± 0.1
0.42 ± 0.2 Ref. (54)
0.5 ± 0.1 Ref. (53)
Twiststretch
coupling
D
(unitless)
11.5 ± 3.3
17 ± 5
17 ± 7 Ref. (54)
22 ± 5 Ref. (53)
21 ± 1 Ref. (59)
Measurement in this work are in TE buffer with 100 mM NaCl. The cited literature
values were all obtained around physiological pH and in 100150 mM monovalent salt.
†Values for the torsional persistence length are extrapolated to high forces using the
MorozNelson model (36, 37) or measured at forces > 15 pN.
18
Table S2: Predictions of the basestep model for the elastic parameters of dsRNA
and dsDNA.
Parameter
Symbol
(units)
dsRNA
dsDNA
2.8_noprot1
2.8_all1
2.8_noprot1
2.8_all1
Bending
persistence
length
A (nm)
66.3
46.9
54.7
39.4
Stretch
modulus
S (pN)
979
776
1956
1504
Torsional
persistence
length
Clim (nm)
53.0
42.4
28.8
40.5
Slope of the
extension vs.
turns response
close to zero
turns
d
Δ
L/dN
(nm/turn)
0.797
0.650
0.473
0.743
Twiststretch
coupling2
D
(unitless)
30.3
19.6
35.9
43.4
1The “2.8_noprot” data set contains nucleic acid crystal structures that have been solved
to a resolution of better or equal to 2.8 Å and do not include bound proteins; the “2.8_all”
data set has the same resolution cut off but includes structures that feature bound
proteins.
2The twist coupling was calculated from the predictions of the slope dΔL/dN and stretch
modulus S for each parameter set using Eqn. 20.
19
Table S3: Changes of average basepair step parameters upon stretching for 100bp
dsDNA and dsRNA helices.
Force (pN)
Avg. extension
(nm)
Avg. rise (Å)1
Avg. roll (°)1
Avg. twist (°)1
D
N
A
1
28.5
3.300 / 0.006
1.56 / 0.001
35.22 / 0.004
5
30.9
3.305 / 0.020
1.56 / 0.007
35.27 / 0.019
10
31.6
3.310 / 0.036
1.57 / 0.005
35.33 / 0.039
20
32.2
3.323 / 0.075
1.56 / 0.007
35.45 / 0.076
40
32.8
3.349 / 0.153
1.55 / 0.009
35.69 / 0.152
R
N
A
1
24.0
3.225 / 0.002
7.85 / 0.008
31.74 / 0.008
5
25.9
3.228 / 0.012
7.75 / 0.027
31.80 / 0.029
10
26.6
3.230 / 0.021
7.62 / 0.055
31.88 / 0.061
20
27.2
3.237 / 0.045
7.36 / 0.108
32.03 / 0.115
40
27.9
3.250 / 0.093
6.90 / 0.201
32.33 / 0.226
Simulations were performed using the “2.8_noprot” parameter set. The changes of shift,
slide and tilt upon stretching are small (below 0.02 standard deviations) and therefore not
shown.
1The first value is the average parameter, followed by the corresponding Zscore.
20
Table S4: Effects of individual parameters in the covariance matrix for dsDNA.
Bending
persistence
length (nm)
Change
(%)
Torsional
persistence
length (nm)
Change
(%)
Link vs.
force slope
(rad/pN)1
Change
(%)
Original
53.0
29.0
0.226
Shift_half2
53.0
0.1
28.7
1.0
0.224
1.1
Shift_double2
53.1
0.1
28.1
3.0
0.209
7.5
Slide_half
53.0
0.0
28.7
1.0
0.246
8.9
Slide_double
53.0
0.0
32.0
10.3
0.200
11.6
Rise_half
53.0
0.0
30.3
4.5
0.239
5.8
Rise_double
53.0
0.1
27.9
3.8
0.227
0.5
Tilt_half
62.8
18.6
28.9
0.5
0.237
4.7
Tilt_double
40.4
23.8
28.3
2.4
0.227
0.5
Roll_half
78.8
48.7
28.2
2.8
0.208
8.0
Roll_double
32.0
39.5
29.9
2.9
0.188
16.7
Twist_half
53.0
0.1
57.6
98.4
0.248
9.8
Twist_double
52.8
0.3
13.3
54.1
0.249
10.3
ShiftSlide_revsign3
52.9
0.1
28.1
3.1
0.228
0.7
ShiftRise_revsign
53.0
0.0
28.4
2.0
0.224
0.8
ShiftTilt_revsign
53.0
0.1
28.9
0.3
0.215
4.9
ShiftRoll_revsign
53.0
0.0
30.9
6.3
0.226
0.1
ShiftTwist_revsign
53.0
0.0
26.5
8.5
0.216
4.7
SlideRise_revsign
53.1
0.2
28.5
1.8
0.170
24.8
SlideTilt_revsign
52.9
0.1
28.8
0.7
0.220
2.7
SlideRoll_revsign
53.0
0.0
31.4
8.2
0.262
15.7
SlideTwist_revsign
53.0
0.1
25.9
10.9
0.177
21.5
RiseTilt_revsign
53.1
0.1
30.0
3.4
0.207
8.3
RiseRoll_revsign
53.0
0.1
29.7
2.3
0.216
4.3
RiseTwist_revsign
53.0
0.0
27.1
6.6
0.226
200.0
TiltRoll_revsign
53.1
0.1
31.0
6.9
0.229
1.4
TiltTwist_revsign
52.9
0.1
28.2
2.8
0.248
9.4
RollTwist_revsign
56.2
6.2
25.4
12.5
0.234
3.4
Calculations are based on the 2.8_noprot parameter set.
1Here we used the slope of linking number (bead rotation) vs. stretching force to evaluate
twiststretch coupling, as this quantity is faster to evaluate in simulations than the slope
of extension vs. turns at constant force.
2Halving or doubling the variance of ‘shift’ parameter in the covariance matrix.
3Reverse the sign of the shiftslide covariance in the covariance matrix.
21
Table S5: Effects of individual parameters in the covariance matrix for dsRNA.
Bending
persistence
length (nm)
Change
(%)
Torsional
Persistence
length (nm)
Change
(%)
Link vs.
force slope
(rad/pN)1
Change
(%)
Original
62.9
52.8
0.174
Shift_half2
63.0
0.1
52.7
0.3
0.164
6.2
Shift_double2
62.9
0.0
58.8
11.3
0.161
7.6
Slide_half
62.9
0.0
54.8
3.7
0.173
0.9
Slide_double
62.9
0.0
52.8
0.1
0.176
0.7
Rise_half
62.9
0.0
54.2
2.5
0.143
17.9
Rise_double
62.9
0.0
53.7
1.6
0.152
12.6
Tilt_half
73.7
17.1
51.3
3.0
0.155
10.9
Tilt_double
48.7
22.6
53.9
2.0
0.172
1.6
Roll_half
92.1
46.3
55.9
5.8
0.237
35.9
Roll_double
38.5
38.7
49.3
6.6
0.047
73.2
Twist_half
64.0
1.7
102.9
94.8
0.122
29.9
Twist_double
60.8
3.3
28.0
47.1
0.316
81.1
ShiftSlide_revsign3
63.0
0.1
54.0
2.3
0.161
7.8
ShiftRise_revsign
63.2
0.4
54.2
2.7
0.174
0.1
ShiftTilt_revsign
63.0
0.1
52.6
0.4
0.151
13.4
ShiftRoll_revsign
62.9
0.0
52.8
0.0
0.154
11.8
ShiftTwist_revsign
62.8
0.1
54.7
3.6
0.151
13.3
SlideRise_revsign
62.9
0.1
53.5
1.3
0.174
0.1
SlideTilt_revsign
62.9
0.1
53.6
1.4
0.191
9.5
SlideRoll_revsign
62.8
0.2
55.5
5.0
0.166
4.5
SlideTwist_revsign
62.9
0.0
54.0
2.3
0.071
59.3
RiseTilt_revsign
62.9
0.0
53.2
0.7
0.167
4.3
RiseRoll_revsign
62.8
0.1
54.0
2.2
0.151
13.6
RiseTwist_revsign
62.9
0.0
54.4
3.0
0.100
42.8
TiltRoll_revsign
62.9
0.1
51.1
3.3
0.179
2.6
TiltTwist_revsign
63.0
0.2
55.0
4.1
0.167
4.0
RollTwist_revsign
68.0
8.0
46.3
12.4
0.048
72.2
Calculation are based on the 2.8_noprot parameter set.
1Here we used the slope of linking number (bead rotation) vs. stretching force to evaluate
twiststretch coupling, as this quantity is faster to evaluate in simulations than the slope
of extension vs. turns at constant force.
2Halving or doubling the variance of ‘shift’ parameter in the covariance matrix.
3Reverse the sign of the shiftslide covariance in the covariance matrix.
22
Table S6. Original and refitted variance and covariance parameter sets for dsDNA
and dsRNA simulations.
DNA_2.8
_all
DNA_2.8
_noprot
DNA_refit
RNA_2.8
_all
RNA_2.8
_noprot
RNA_refit
shift1
0.64
0.58
0.58
0.66
0.57
0.57
slide1
0.82
0.86
0.86
0.44
0.39
0.39
rise1
0.25
0.23
0.33
0.24
0.20
0.29
tilt1
3.85
3.58
3.58
3.52
2.87
2.87
roll1
6.25
5.19
5.92
5.2
4.33
4.94
twist1
5.43
6.27
3.14
4.73
4.25
2.69
shiftslide2
0.02
0.02
0.02
0.01
0.03
0.03
shiftrise2
0.02
0.03
0.02
0.01
0.01
0.01
shifttilt2
0.34
0.27
0.27
0.25
0.36
0.36
shiftroll2
0
0.01
0.01
0.01
0.02
0.02
shifttwist2
0.01
0.04
0.08
0.03
0.01
0.01
sliderise2
0.07
0.11
0.08
0.23
0.19
0.13
slidetilt2
0.01
0.01
0.01
0.07
0.00
0.00
slideroll2
0.17
0.09
0.08
0.13
0.10
0.08
slidetwist2
0.4
0.25
0.49
0.39
0.40
0.64
risetilt2
0.01
0.05
0.03
0.03
0.03
0.02
riseroll2
0.03
0.16
0.10
0.23
0.17
0.11
risetwist2
0.26
0.37
0.16
0.1
0.11
0.12
tiltroll2
0.02
0.01
0.01
0.01
0.01
0.01
tilttwist2
0
0.02
0.05
0.07
0.02
0.02
rolltwist2
0.42
0.42
0.73
0.1
0.13
0.09
1The standard deviation of shift, slide, rise, tilt, roll, twist. Units for shift and slide and
rise are Å and units for tilt, roll and twist are degree.
2The correlation coefficient (unitless) for shift vs. slide and so on.
23
Table S7: Mechanical properties from the modified parameter sets.
Parameter
Symbol
(units)
dsRNA
dsDNA
2.8_noprot
modified
2.8_noprot
modified
Bending
persistence
length
A (nm)
66.3
59.9
54.7
47.0
Stretch
modulus
S (pN)
979
685
1956
1067
Torsional
persistence
length
Clim (nm)
53.0
103.7
28.8
124.8
Slope of the
extension vs.
turns response
close to zero
turns
d
Δ
L/dN
(nm/turn)
0.797
0.832
0.473
0.599
Twiststretch
coupling
D
(unitless)
30.3
22.1
35.9
24.8
24
SUPPLEMENTARY FIGURES
Fig. S1. Force calibration and stretching experiments on dsRNA and dsDNA in the
magnetic tweezers.
a) Force calibration including spectral corrections. Comparison of different methods
to determine the applied stretching forces in the MT from the position fluctuations of the
magnetic beads. Data were obtained using MyOne beads and 4.2 kbp dsRNA tethers with
a set of verticallyoriented magnets with a 1 mm gap. Points are the mean and standard
deviation from 16 independent tethers. The same experimental data were analyzed using
three different methods to determine the stretching forces (see also the “Force calibration
in the magnetic tweezers” section): 1) by fitting of the power spectral density using the
25
method by Lansdorp and Saleh (11) (green diamonds), 2) from an analysis of the Allan
variance of the data (11) (blue squares) , and 3) by fitting the integral of the power
spectral density of the data (10) (red circles). The three methods yield identical results,
within experimental error. Note that symbols partially overlap. All force values reported
below were obtained by fitting the integral of the power spectral density.
b) Force calibration for MyOne beads. The applied stretching force as a function of the
distance Zmag between a set of verticallyoriented magnets with a 1 mm gap and the flow
cell surface for MyOne beads. Data for 4.2 kbp dsRNA tethers (red symbols; points are
the mean and standard deviation
σ
from 16 independent tethers) agree within
experimental error with data obtained previously (2) using 20.6 kbp DNA tethers (blue
circles). The RNA data were fit to an empirical double exponential model of the form
F