We studied the thermal fluctuations of single DNA molecules with a novel optical tweezer based force spectroscopy technique. This technique combines femtonewton sensitivity with millisecond time resolution, surpassing the sensitivity of previous force measurements in aqueous solution with comparable bandwidth by a hundredfold. Our data resolve long-standing questions concerning internal hydrodynamics of the polymer and anisotropy in the molecular relaxation times and friction coefficients. The dynamics at high extension show interesting nonlinear behavior

Meiners, Jens-Christian

Quake, Stephen R.

Caltech Authors - Main

VOLUME 84, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 22 MAY 2000
5014Femtonewton Force Spectroscopy of Single Extended DNA Molecules
Jens-Christian Meiners and Stephen R. Quake
Department of Applied Physics, California Institute of Technology, Pasadena, California 91125
(Received 27 December 1999)
We studied the thermal fluctuations of single DNA molecules with a novel optical tweezer based force
spectroscopy technique. This technique combines femtonewton sensitivity with millisecond time reso-
lution, surpassing the sensitivity of previous force measurements in aqueous solution with comparable
bandwidth by a hundredfold. Our data resolve long-standing questions concerning internal hydrody-
namics of the polymer and anisotropy in the molecular relaxation times and friction coefficients. The
dynamics at high extension show interesting nonlinear behavior.
PACS numbers: 87.15.Ya, 05.40.Jc, 36.20.EySingle molecule force spectroscopy has become a valu-
able tool to study the intramolecular forces involved in un-
folding a protein [1,2] or inducing conformational changes
in a polymer [3]. These chemical forces are typically of
the order of tens of piconewtons and can readily be mea-
sured with atomic force microscope techniques. Entropic
forces as low as 100 fN have been measured with opti-
cal and magnetic traps [4–6], but thermal motion of the
force-transducing latex bead in the trap limits the time
resolution of these techniques and allows dynamic mea-
surements only for piconewton force fluctuations [7,8].
We have developed a scheme using dual optical tweezer
force sensors that overcomes this limitation by making
a cross-correlated heterodyne measurement. The thermal
motions of the two beads are independent and uncorre-
lated, allowing us to measure the correlations introduced
by a molecule whose ends are attached to the beads. Thus,
one can measure the dynamic motion of the molecule with
high temporal and force resolution by computing the cross
correlation between the measured forces acting on the
two beads. This femtonewton force spectroscopy (FFS)
scheme is general and can in principle be adapted to make
high-resolution measurements of the average power stroke
of molecular motors such as myosin, kinesin, and RNA
polymerase. It is superior to measuring individual events
[8,9] or the instantaneous cross correlation [10] since it
effectively self-averages over the entire period of the mea-
surement, giving a concomitant improvement in the signal-
to-noise ratio and removing human bias in separating true
events from noise fluctuations.
We used FFS to study the thermal fluctuations of a single
DNA molecule. DNA has proved to be a useful model sys-
tem to study the complex dynamic behavior of polymers,
an intriguing subject of long-standing theoretical and ex-
perimental interest. An important application of polymer
dynamics has been to understand the behavior of polymers
in hydrodynamic flow, where one observes fascinating ef-
fects such as drag reduction in turbulent pipe flow. To help
understand this problem, many years ago De Gennes pre-
dicted that polymers would exhibit a “coil stretch” phase
transition in extensional flow [11]. Theoretical arguments
and experiments have shown that the critical fluid strain0031-90070084(21)5014(4)$15.00rate at which polymers extend is ´  1t, where t is the
fundamental relaxation time of the polymer. Subtle hys-
teresis effects are possible according to how t depends on
extension.
In addition to the practical reasons to investigate
extended polymer dynamics, there are also fundamental
questions about the equilibrium dynamics of a polymer.
For example, it has been predicted that due to symmetry
breaking t will “split” into distinct longitudinal and
transverse components as the polymer is extended. Two
potential contributions have been identified: geometric
effects due to the nonlinear spring constant [12], and an
alignment effect due to anisotropic friction coefficients
[13,14]. Another historically important example is the
notion that internal hydrodynamic effects should cause
the polymer’s friction coefficient to increase as it extends,
causing it to go from a “non-free-draining” state to a
“free-draining” state. It has been suggested that this
would be the dominant effect on t, which would then
also increase [11,13]. However, there is little direct
experimental evidence about t or even the behavior of
the friction coefficient. One study of DNA is extensional
flow provided indirect evidence that the molecule does
not become free draining up to 80% extension [15], but
subsequent theoretical interpretations of the experiment
are conflicting [16–19]. Another study with DNA ex-
tended up to 80% with optical tweezers showed that the
internal modes are related by a power law whose exponent
is an intermediate value between the free-draining and
non-free-draining predictions [20]. The data from that
study also show that the nonlinearity of the polymer
spring constant affects the transverse relaxation time [12].
Finally, it has been shown that when DNA is confined
to a slit geometry, the walls screen the hydrodynamic
interactions and the polymer becomes free draining [21].
We studied the behavior of 6 DNA molecules at a total
of 36 different extensions ranging from 74% to 92%. The
experimental system is shown schematically in Fig. 1(a).
A l-phage DNA molecule with beads attached to its ends
is stretched with a dual-beam optical tweezer apparatus.
We record the Brownian motion of the beads by imaging
the scattered laser light from each sphere onto a quadrant© 2000 The American Physical Society
VOLUME 84, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 22 MAY 2000FIG. 1. Measuring the thermal fluctuations of a single DNA
molecule with optical tweezers. (a) A DNA molecule of the
l-phage with beads attached to its ends is stretched with a dual-
beam optical tweezer apparatus. We record the Brownian mo-
tion of the beads by imaging the scattered laser light from each
sphere onto a quadrant photodiode. This gives a measurement of
the displacement of the bead from the focus of the laser beam in
the x and y direction. (b) The dynamic model used to find the
spring constants and friction coefficients of the DNA molecules
from the measured correlation functions. The DNA is modeled
as a sphere with a friction coefficient zDNA and two springs,
each of strength 2 kDNA, giving an overall spring constant
of kDNA.
photodiode. This gives a measurement of the displacement
of the bead from the focus of the laser beam in the x and
y directions. In order to minimize the cross talk between
the signals from the two traps, the trapping beams are
orthogonally polarized and chopped alternatingly at a
frequency of 80 kHz, and the scattered light is detected
synchronously in the appropriate polarization. For each
data set, 107 such position measurements were made at
a rate of 20 kHz for each trap. A typical trap relaxation
time was ttrap  0.3 ms, which corresponds to a spring
constant of ktrap  25 pNmm. Details of the experimen-
tal apparatus have been published elsewhere [22]. The
DNA-bead complexes were prepared by labeling l-DNA
at either end with biotin and digoxigenin (dig), respec-
tively [23]. Monoclonal dig antibodies from mouse cells
(Roche) were bound to the dig-labeled DNA end. This
DNA-antibody complex was incubated over night with
equal amounts of goat antimouse IgG-coated (where IgG
denotes immunoglobuling in type G) beads (polysciences)
and streptavidin-coated beads (Bangs Laboratories), both
1 mm in diameter. A threefold excess of DNA over
beads was used in order to compensate for incomplete
labeling and binding. Finally, the bead-DNA-bead com-
plexes were diluted in 10 mM NaCl-TE buffer (10 mM
TrisCl, 1 mM EDTA, 10 mM NaCl, pH 8.0) to a volume
fraction of f  1026. This solution was hermetically
sealed in a 100 mm deep sample cell. Measurements
were taken while the DNA was held at a depth of
15 mm, and the absolute distance between the centers
of the beads was measured for each data set by video
microscopy.
From this time-resolved measurement of the forces act-
ing onto the beads the correlation functions  f1tf10, f2tf20, and  f1tf20 were computed. The auto-
correlation functions fit an exponential
 fitfi0  kakBTe2tta , (1)
yielding the spring constant ka and time constant ta of
the relaxation of the beads in their traps. Similarly, the
end-to-end cross-correlation function of the polymer fluc-
tuations can be described by a stretched exponential
 f1tf20  kckBTe2ttc
n
, (2)
where kc is a spring constant, tc a relaxation time, and n
a stretch exponent [24]. Hydrodynamic coupling between
the beads contributes an additional fast-decaying term to
the correlation function, which is well understood from
earlier work [22]. This hydrodynamic interaction gives
rise to the pronounced dip that is seen in the correlation
functions (Fig. 2) during the first millisecond for smaller
extensions. In the absence of the DNA molecule it is
described by
ht  e211´itt 2 e212´itt , (3)
with ´x  3a2D longitudinally and ´y,z  3a2D in the
transverse directions, where a and D denote the radius and
the separation of the beads, respectively. The presence of
the DNA screens this hydrodynamic coupling between the
end spheres to about 60% of its native amplitude.
The measured relaxation times and spring constants are
shown in Figs. 3(a) and 3(b). We fixed the exponent
FIG. 2. Relaxation of a single, extended DNA molecule as re-
vealed by the cross correlation between the forces acting onto
the end beads. Panels (a) and (b) show the longitudinal re-
laxation of a molecule that is extended to 92% and 82% of
its contour length, respectively. Only every third data point is
plotted. Panels (c) and (d) show the transverse relaxation at
92% and 88%, respectively, with every tenth data point being
plotted in the graph. A stretched single exponential decay has
been fitted to all data sets, with an additional correction for the
hydrodynamic interactions between the beads. This hydrody-
namic interaction gives rise to the pronounced dip that is seen in
the correlation functions during the first millisecond for smaller
extensions. The ultimate force sensitivity of the apparatus is
6 fN rms.5015
VOLUME 84, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 22 MAY 2000FIG. 3. Thermal fluctuations of a DNA molecule as a function
of its extension, compiled from measurements on six different
molecules. Experimental data points in the longitudinal direc-
tion are indicated as circles, and square mark the transverse
data points. Uncorrected data from the fits are shown as open
symbols, and the corrected data points are shown as the filled
symbols. (a) The relaxation times of the DNA-dumbbell as a
function of extension as determined from the least-square fits to
Eq. (2) as well as the corrected relaxation times from Eq. (4).
The solid lines indicate the predictions from the simple modi-
fied Rouse model [12] in Eq. (5). (b) The longitudinal and
transverse spring constants of the molecule, as determined from
the amplitude of the fluctuations, as well as the corrected val-
ues accounting for the finite stiffness of the optical traps. The
predictions of the wormlike chain model for both directions are
indicated as solid lines. (c) The stretch exponent n for the ther-
mal relaxation of the polymer. The fitted values of n are shown
above 82% extension. Below 82%, it was not necessary to let
n be a free parameter in the fits. (d) The friction coefficient of
the DNA molecule as a function of its extension.5016n  1 when fitting the data at lower extensions. However,
above 82% there were clear systematic errors in the fits,
and the simple exponential decay deviated from the data
both at short and long time scales. This behavior does
not appear to be due to higher order Rouse-Zimm modes,
since those modes would be even more evident at lower ex-
tension where the time resolution is better. Nevertheless,
we modified Eq. (2) to include various models of higher
order mode structures with a single new parameter, and
found that the fits still exhibited systematic errors. Adding
enough free parameters led to good fits, but the values of
the fitted parameters were inconsistent with known [20]
or expected [24] features of the mode structure. We then
returned to Eq. (2) and allowed n to be fit as a free parame-
ter, and found that the fits were free of obvious system-
atic errors. The fitted parameter n increased from 1.0 to
1.6 as the extension increased [Fig. 3(c)]. While such su-
perexponential relaxation is well known in Levi-flight-like
anomalous diffusion problems [25], we can only speculate
about its physical origin in the present situation. Possible
causes might include mode mixing in the polymer, per-
haps due to sequence heterogeneity or the highly nonlinear
spring constant, or complex dynamic effects due to transi-
tion force-assisted denaturation.
If ktrap ¿ kDNA and ttrap ø tDNA, then the fitted pa-
rameters from above are kc  kDNA, tc  tDNA, ka 
ktrap , and ta  ttrap . However, as either the spring con-
stant or relaxation time of the DNA approaches that of the
trap, there will be mixing in the measured values from the
cross- and autocorrelations. It is straightforward to solve
the dumbbell-like model system outlined in Fig. 1(b) and
show that the corrections are
kDNA  kc
zbead
taka 1 kc
,
tDNA  tc
ka1 2 tatc 2 kc1 1 tatc
ka 1 kc 1 2 tatc
.
(4)
These corrections are negligible at the smallest exten-
sions, but become significant when the spring constant
of the DNA increases. For comparison both the raw,
uncorrected data and the corrected values for the relax-
ation times and spring constants are shown in Figs. 3(a)
and 3(b), along with the predicted values from theo-
retical calculations. The uncorrected as well as the
corrected data show a clear trend towards higher spring
constants and shorter relaxation times as the extension
increases, as well as a marked anisotropy. In fact, the
corrected longitudinal spring constant derived from
Eqs. (1), (2), and (4) increases rapidly from 0.2 pNmm
at 74% extension to 4.7 pNmm at 92% extension
[Fig. 3(b)], while the transverse spring constant reaches
only 0.3 pNmm at 92% extension. This behavior is in
good agreement with the predictions from the wormlike
chain model [12], which yields for the longitudinal
and transverse spring constants of an extended polymer
kDNA,k  dFdxjE and kDNA,  FEE, where Fx
VOLUME 84, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 22 MAY 2000is the force-extension relationship of a wormlike chain
[18], Fx  kBTlp  14 1 2 xL022 2
1
4 1 xL0,
and E the end-to-end distance of the polymer. Using a
persistence length lp of 53 nm, a least-square fit for the
longitudinal spring constant indicates a contour length
L0 of approximately 16.1 6 0.2 mm, which is somewhat
below the crystallographic contour length of the DNA
molecule of 16.4 mm. This 2% discrepancy might be
caused by steric effects at the anchor points of the DNA
as well as uncertainties in the calibration of the video
microscope and the bead diameter.
The relaxation times of the DNA are remarkably well
described by the same model if the longitudinal and trans-
verse friction coefficients of the DNA are assumed to be
those of a rigid rod with the dimensions of the molecule,
i.e., a length of 16.4 mm and a diameter d of 2.5 nm.
Those relaxation times are
tDNA,k 
2hL0lp
pkBT lnL0d
∑
1
2L01 2 EL03
1
1
L0
∏21
,
tDNA, 
4hL0lp
pkBT lnL0d
(5)
3
∑
1
4E1 2 EL02
2
1
4E
1
1
L0
∏21
,
and the comparison with the experimental data is shown in
Fig. 3(a).
One can use the measured values of the relaxation time
and spring constant to derive the DNA molecule’s fric-
tion coefficient, since tDNA ~ zDNAkDNA, where zDNA
denotes the dynamic friction coefficient of the polymer
chain and will take on distinct values for the longitudi-
nal and transverse directions. A proportionality factor is
needed to ensure that this definition of zDNA is consistent
with the fluctuation-dissipation theorem [24]. Its numeri-
cal value is 0.25 in the dumbbell-like model [Fig. 1(b)],
and 1p2 in a continuum model. The measured friction
coefficients are independent of extension over the mea-
sured range [Fig. 3(d)] and in the longitudinal direction
zDNA,k  7.60.2 3 1029 N sm, while in the transverse
direction zDNA,  17.30.8 3 1029 N sm. The ratio of
transverse to longitudinal is 2.28 (0.12), close to the ex-
pected value of 2 and clearly distinct from unity. The given
errors are statistical and do not include uncertainties in the
numerical prefactor in our dynamic model, which can at
least in part account for the fact that the measured abso-
lute values are 25% and 15% smaller than the predicted
values of the friction coefficients of a rigid rod whose di-
mensions are comparable to the DNA molecule.
In conclusion, we have used FFS to study the dynamics
of an extended polymer and measured the anisotropy of
the longitudinal and transverse relaxation times. We show
that this effect is due to anisotropy in both the friction co-
efficient and spring constant, from hydrodynamic and geo-metric effects, respectively. The anisotropy in the friction
coefficient is consistent with rodlike (or freely draining)
hydrodynamic behavior. The fact that the fundamental re-
laxation time t decreases as a function of extension shows
that hysteresis is unlikely in the coil-stretch transition, and
will also serve to broaden the transition. The correlation
functions show superexponential relaxation at high exten-
sions, which may indicate the presence of interesting new
physics.
This work was supported by the National Science Foun-
dation CAREER program.
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Femtonewton Force Spectroscopy of Single Extended DNA Molecules

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Femtonewton Force Spectroscopy of Single Extended DNA Molecules

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