Biophysical Journal
Volume 72
Stretching
March 1997
1335-1346
1335
DNA with Optical Tweezers
Michelle D. Wang,* Hong Yin,# Robert Landick,§ Jeff Gelles,# and Steven M. Block*
'Department of Molecular Biology, Princeton University, Princeton, New Jersey 08544; #Department of Biochemistry, Brandeis University,
Waltham, Massachusetts 02254; and §Department of Bacteriology, University of Wisconsin, Madison, Wisconsin 53706 USA
ABSTRACT
Force-extension
(F-x) relationships were measured for single molecules of DNA under a variety of buffer
conditions, using an optical trapping interferometer modified to incorporate feedback control. One end of a single DNA
molecule was fixed to a coverglass surface by means of a stalled RNA polymerase complex. The other end was linked to a
microscopic bead, which was captured and held in an optical trap. The DNA was subsequently stretched by moving the
coverglass with respect to the trap using a piezo-driven stage, while the position of the bead was recorded at nanometerscale resolution. An electronic feedback circuit was activated to prevent bead movement beyond a preset clamping point by
modulating the light intensity, altering the trap stiffness dynamically. This arrangement permits rapid determination of the F-x
relationship for individual DNA molecules as short as ~1 ILm with unprecedented accuracy, subjected to both low (~0.1 pN)
and high (~50 pN) loads: complete data sets are acquired in under a minute. Experimental F-x relationships were fit over
much of their range by entropic elasticity theories based on worm-like chain models. Fits yielded a persistence length, Lp, of
~47 nm in a buffer containing 10 mM Na+. Multivalent cations, such as Mg2+ or spermidine3+, reduced Lp to -40 nm.
Although multivalent ions shield most of the negative charges on the DNA backbone, they did not further reduce Lp
significantly, suggesting that the intrinsic persistence length remains close to 40 nm. An elasticity theory incorporating both
enthalpic and entropic contributions to stiffness fit the experimental results extremely well throughout the full range of
extensions and returned an elastic modulus of -1100 pN.
INTRODUCTION
The mechanical flexibility
of DNA plays a key role in all of
its cellular functions, including folding, packaging, regula-
spects, our completed instrument resembles feedback-enhanced optical traps recently developed for work with the
tion, recombination,
replication, and transcription. The intimate connections between DNA's molecular interactions
and its physical properties are therefore of considerable
interest to biologists (Yang et al., 1995). Moreover, free
molecules of DNA in solution are relatively well-behaved,
and therefore afford an excellent experimental system for
studies of polymer physics (Smith et al., 1992; Perkins et
actomyosin system (Simmons et al., 1996; Molloy et al.,
1995), except that the feedback mechanism is implemented
by modulating the light intensity instead of the position of
the trapping beam. This results in a simpler system with a
distinct set of advantages and disadvantages, discussed
later. The optical layout for the instrument is shown in
Fig. 1 A.
al., 1994a,b, 1995; Smith et al., 1996; Cluzel et al., 1996).
Our own interest in the micromechanical properties of DNA
was sparked by recent work monitoring
transcription
of
DNA directed by single molecules of RNA polymerase in
The extensibility of DNA has been measured using forces
applied by magnets or fluid flow (Smith et al., 1992), and
more recently with optical traps (Smith et al., 1996) or glass
microneedles (Cluzel et al., 1996). To an excellent approx-
vitro (Schafer et al., 1991; Yin et al., 1994, 1995). For such
assays, the series elastic compliance of individual
DNA
molecules must be ascertained and factored into computations of polymerase translocation. We were therefore motivated to develop an improved means of rapidly determining
this linkage compliance using the optical trapping system
itself, for molecules with lengths around -1 ILm-more
imation, single molecules of DNA behave like ideal, entropic springs for low-to-moderate
extension beyond their
rest length. The elastic behavior of such springs is characterized by a polymer persistence length, 4. Data fits to the
low-force regime of the F-x curve gave persistence lengths
of -53 nm (in 10 roM Na+ buffer; Bustamante et al., 1994)
or -15 nm (in 80 roM Na+ buffer, Cluzel et al., 1996).
than an order of magnitude shorter than sizes used in earlier
Comparisons
measurements
with the pre-
single-molecule
dictions of entropic theory have been limited
mainly to the
studies of DNA
elasticity.
In certain re-
of experimental
Receivedfor publication 7 October 1996 and in final form 6 December
low-force regime, where the theory is most applicable. Once
drawn out to their full contour lengths, purely entropic
springs become inextensible. Molecules of DNA, however,
1996.
display
additional
1996)
the on gl n
.
Address repnnt requests to Dr. Steven M. Block, 316 Schultz Lab, DefM
partment
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ar
.
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ogy,
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.
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,
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compliance
of
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W
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IS
this
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limit
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(Smith
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et al.,
can
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Tel.: 609-258-2638; Fax: 609-258-1035; E-mail: block@watson.
princeton.edu.
tenzed by an elastIc modulus, Ko. Recently, elastIcIty theories have been extended to account for both entropic and
Dr. Yin's presentaddressis Departmentof Molecular and Cellular Biology, Harvard University, Cambridge,MA 02138.
@ 1997by the Biophysical Society
enthalpic contributions to stretch (Marko and Siggia, 1996;
Odijk, 1995). The revised expressions permit a fairly complete comparison with experimental measurements over the
0006-3495/97/03/1335/12 $2.00
full
range of extensions,
and make possible
the simulta-
Wang et al.
Stretching DNA with Optical Tweezers
1337
(NTPs) are not presentin thesebuffers, so transcriptionalelongationdoes
not occur; the enzymecomplex merely servesas a convenientway to fix
one end
DNA molecule to the coverglass.During a stretching
. of the
h b d . h ld . .
hil th stageIS
. move
d back
expenment,
t e ea IS e InSIde the trapwee
where Kx is the trap stiffness in the x direction and (3 is the
viscous drag coefficient.
Very near the coverglass, this
andforthwith a periodicwavefonnvoltageappliedto a piezoelectric
stage,
As thepolymerase
is movedawayfromthetrapposition,theDNA tether
is stretched,
pullingthebeadfromthecenterof thetrapandincreasing
the
restoringforce. Oncethe beadpositionreachesthe clamp set point.
of the surface to the bead, because of the strong dependence
of drag on height, h (Faxen' s law; see Svoboda and Block,
1994b). For example, drag doubles at hlr "" 1.2, where r is
the bead radius: the bead is therefore about a radius away
feedback is activated, increasing the laser intensity to hold bead position
..
constant while the DNA tether continues to stretch.
fr om the SWlace
c
when the corner f requency dr0pS by a
The relationshipbetweenstagedisplacementand piezo control voltage
was established by tracking beads immobilized
on a coverglass surface,
using a video-based,centroid cross-correlationmethod accurateto the
subpixel level (Gelles et al., 1988). Distance was establishedusing a
diamond-ruledgrid with 10 IJ-mspacing,traceableto the National Bureau
of Standards(Svobodaet al., 1993).
f requencybecomesexceed.mg1y sensItIve
. . to the proxIllllty
'.
factor of -2.
This method was used to set the trap center to
one radius over the specimen plane, with an accuracy of
:t50 nm for a 0.52-J1,m-diameter bead.
Trap height determination
CALIBRATION
T
"t
rap
pOSI
METHODS
"
Ion
t
con
I
ro
The trap centeris defined asthe equilibrium position of the
centerof an optically trappedspherewith no externalload
(Fig. 1 B). Becauseof scatteringforces,this point doesnot
coincide with the laser beam waist (focus) and will generally be locatedin a planebeyondthe waist, at a distancethat
dependsupon the size of the sphereand its relative refractive index. The trap height, Ztrap'is defined as the vertical
distance between the trap center and the surface of the
coverglass,and is affectedby the microscopefocus. Accurate control of this height is essentialto position and stiffnesscalibration, as well as to establishinga reproducible
experimentalgeometryin DNA stretchingexperiments.In
t
( d all th
t.
our sys em an
0 er op ICal traps), ad.~ust.mg the llll-.
croscope's focal control moves both the position of the
specimenplane(theposition whereobjectsarein focus) and
the position of the laserfocal plane (hencethe trap center).
The distancebetweenthe trap center and specimenplane
can be controlled independentlyby meansof the z-control
of an externaltelescope,which adjuststhe convergenceof
the laser light without alterationsto the microscopefocus
(Svobodaand Block, 1994b;seealso Fig. 1 A). Trap height
was thereforeestablishedin two steps.First, an offset between the trap center and the specimenplane was fixed
using the externalz-control. Next, the trap height was adjusted by refocusing the microscopeobjective by a preset
amount.
Once the distance between trap center and the objective
focus
was
established,
the
trap
height
was
preset
by
refo-
h b..
cusmgt e 0 ~ectIvea small distanceabove the coverg1ass
surface.A long indicator arm (-30 cm) was constructed
and attachedto the fine-focus knob of the microscope;this
permitted us to turn the control through a small, reproducible angle, with a focal accuracy of better than 80 nm.
Becauseof the refractive index mismatch at the samplecoverglassinterface,the changein focal height is lessthan
the physical movementof the objective lens, by a factor of
-1.2 (Wiersma and Visser, 1996; Hell et al., 1993); this
effect was included in estimatingthe trap height.
.
POSI
ot O
Ion d et ec t or an d t rap St oIft ness ca l oIb rat oIon
The interferometerprovides an analog voltage output, Vd,
from a normalizeddifferential amplifier encodingdisplacement. This output is most sensitive to motion along the
shear direction of the Wollaston prism in the specimen
plane (x direction) and is relatively insensitiveto displacementsin the y andz directions(unpublisheddata;Denk and
Webb, 1990; Svobodaand Block, 1994a,b).For small displacements,Vd is linear in xbead'henceXbead
= Vdls, where
s is the sensitivity of the interferometer.Earlier determinations of xbeadas a function of Vd had beenconductedwith
silica beads stuck to the coverglasssurface and the trap
heightadjustedto its positionof maximumsensitivity,using
calibrated stage movements(unpublished data; Svoboda
and Block, 1994a).
7i
+
d +
. +'
Three complementarymethodsof trap stiffness calibrarap cen,er e,ermlna,lon
'.
tIon have beendescnbed(Svobodaand Block, 1994b):
To fix the location of the trap centerrelativeto the specimen
1. Power spectralestimation.This involves determining
plane,the microscopeobjectivewas focuseddirectly on the
the cornerfrequencyof the Brownianmotion of the bead,fc,
surface of a clean coverglass,as observedby video-enwhich dependsonly on the ratio of the stiffness to the
hanceddifferential interferencecontrast(VE-DIC) microsviscousdamping.
copy. Then a freely diffusing beadin buffer was captured
2. Equipartition theorem. The stiffness is obtained diand moved vertically toward the surfaceof the coverglass rectly from V2Kx(~ead)
= VzkBT,
wherekBTisBoltzmann's
by using the externalz-control only, while the power spec- constanttimes the absolutetemperature.This involves meatrum of the position signalwascontinuouslymonitored.The
surementof the mean-squaredisplacementof the thermal
power spectrumfor Brownian motion in a harmonicpotenmotion of a trappedbead,(X~ead)'
using a position-calibrated
tial is Lorentzian, with a corner frequencyfc = Kx/21T(3, detectorwith sufficiently high bandwidth and low noise.
I
"
1338
!
Biophysical Journal
Volume 72
March 1997
3. Periodic (e.g., sine wave or triangle wave) oscillation.
This usesviscousdrag forces on a trappedbeadappliedby
oscillating the fluid periodically. For sinusoidal motion,
Kx = 21T/3ffluid(Xfluid/Xbead) when 21T/3ffluid/Kx « 1, where
ffluid and Xfluidare the frequencyand amplitudeof the fluid
movement,and Xbeadis the amplitude of bead movement.
This methodis valid over the linear region of the trap, where
surface,by translatingthe samplethrough calibratedx-displacementsusing the piezo stage;s(r, 0) obtainedthis way
agreed well with extrapolation to h = r by the former
approach.This control also shows that the interferometer
response,Vd, remainslinear with beaddisplacementout to
-130 nm in the x direction from the trap center.
As a further control (and consistencycheck), method 3
the stiffness is independent of displacement; by changing
was used to determine trap stiffness at h
the amplitudeor frequencyof the oscillation, this rangecan
be determined.
Method 1, unlike methods 2 and 3, does not require
absolute calibration of the displacementin terms of the
position signal; it is only necessarythat thesetwo remain
proportional. Methods 1 and 3 require knowledge of the
required prior distance calibration of the interferometer,
which was performedusing methodsdescribedabove.The
stiffnessobtainedin this way agreedwell with that obtained
from the power spectrumalone, confirming that the interferometer sensitivity had been properly established.This
control further showsthat trap stiffnessremainsconstantout
viscousdrag coefficient/3. Whenbeadsare closeto the
surface,small errorsin h lead to large errorsin /3, and
to
~
=
1 .urn. This
140nm in thex directionfrom thetrapcenter.
therefore Kx. Method 1 or 3 is thereforebest suited when
h > 2r.
Calibrationbelow trap height
Calibrationat trap height
For the generalcasewherethe beadis locatedat anarbitrary
height and at an arbitrary z offset from the trap center,s =
We noted significant differencesbetweenthe polystyrene
beadsusedfor theseexperimentsand the silica beadsused
for earlier work (Svobodaand Block, 1993, 1994a),attributable to the high refractive index (hence greater light
scattering)of polystyrene(n 1.6). Becauseof the greater
offset betweenthe beamwaist and trap centerfor polysty"'"
s(h, Zbead).
This interferometer sensitivity was computed
from s(h,O)by scaling from measurementsmade at one
other calibratedlocation, accordingto
(h
) = (h O)~~
s ,Zbead s,
s(r,O).
rene, the interferometersensitivity in the plane of the trap
center was lower than its maximum sensitivity, which occurs near the beam waist; for silica beads,in contrast,the
sensitivities are comparable.Therefore,both s and Kx become functions not only of the height, h, but also of any
wheres(r, Zbead)
is the interferometersensitivitydetermined
at a height equalto the beadradius,h = r, usingbeadsstuck
to the surface. The sensitivity was then found through
calibrated stagetranslationsat different trap heights.This
axial displacement
from the trap center,Zbead(Fig. 1 B).
methodassumes
thatthesensitivity,s, for arbitraryh obeys
Suchdisplacementsarise,for example,whenexternalforces
draw the beadfrom its equilibrium position. In contrast,for
silica beads,neither s nor Kx varies significantly with h or
zbead'Becauseof such differences,earlier methodsof calibration had to be modified, and we adopted the fairly
involved proceduredescribedbelow. In the notation that
follows, s = s(h, Zbead)
and Kx = Kx(h, Zbead)'
where the
argumentsrefer to the height and vertical offset from the
trap center,respectively.
Calibrationof the interferometersensitivityandtrap stiffnessfor a trappedbead was accomplishedas follows. At a
height sufficient to accuratelydetermine /3 (h > 2r), the
positional power spectrumand variance were determined.
The stiffness Kx(h, 0) was computedfrom the comer frequency, and s(h, 0) was computedfrom the equipartition
theorem.BecauseactualDNA stretchingexperimentswere
conductedat h < 2r, where the comer frequencybecomes
less reliable, s and Kx were obtainedby extrapolationfrom
a seriesof calibrationsdone over the rangeh = (600-1600
nm), using the method just described.Within this range,
s-l(h, 0) and Kx(h,0) were nearly linear in h: s(h, 0) and
Kx(h,0) eachdecreasedby -15% as h increased.
As an internal check on the abovemethod,a determination of s(r, 0) was carried out with beads stuck to the
scaling behavior similar to that establishedfor h = r, a
reasonableapproximationwhen hand r are the sameorder
of magnitude. In these experiments,s(r, Zbead)
typically
differed from s(r, 0) by 7-10% under the conditions of
study.The s(r, Zbead)
calibrationshowedthe trap centerto be
located-290 nm abovethe planeof maximuminterferometer sensitivity.Assumingthat this planecorrespondsto the
location of the laser beamwaist, we concludethat the trap
centerlies -290 nm from the beamwaist for thesebeads.
Trap stiffness Kx(h,Zbead)
was taken to equal Kx(h,0). The
error in this approximationwasestimatedto be <6%, based
on laser beam profile calculations (Visser and Wiersma,
1992).
G
t d t
. t.
eomery e ermlnaIon
Determination of the F-x relationship requires measurements of both force and extensionin the direction of the
applied stretch,which lies at a variable angle (Jin the x-z
planeand changesduring the courseof an experiment(Fig.
1 B). Theseparameterswere derivedthrough their geometrical relationshipto the interferometersignalsin a seriesof
four steps,as follows:
1340
Biophysical Journal
Volume 72
March 1997
object fixed via modulation of the optical restoring force
(Fig. 2). In the other devices,the force is alteredby deflecting the laser beam in a direction opposite that in which
displacementoccurs. This effectively stiffens the trap by
moving the trapped object into a steeperregion of the
optical potential (Finer et al., 1994; Simmonset al., 1996).
To do so, two orthogonalacoustoopticdefi.ectors(ADDs)
are used in conjunction with separateposition-feedback
circuits, one each for x and y deflections. In our system,
force is insteadchangedby modulatingthe stiffnessof the
trap, through dynamicadjustmentsto the light level. This is
accomplishedwith a single ADM that affectstrap stiffness
in both x andy directionsequally. Thesedesigndifferences
impart both advantagesand drawbacks.First, the system
here is optically simpler and more economical to build:
ADMs and associateddrivers are less costly than ADDs,
and only a single modulator and feedbackcircuit are required. Second, this schemeis well suited for use with
one-dimensionalposition sensors(e.g., the interferometer),
whereas the other schemeis better suited for use with
two-dimensionalsensors(e.g., quadrantphotodiodedetectors). Third, with the intensity-modulatedclamp, the light
level remains low until it is necessaryto increaseit; this
therefore the clamp functions best at some finite tension.
(However, beam-deflection systems are rarely used to
clamp near the origin either, becausepretensioningis usually desiredto reducethermal noise.)Moreover, the clamp
is unidirectional,in the sensethat the restoringforce always
points toward the center of the trap. In practice, these
limitations have not proved troublesomefor this work.
Fig. 3 illustrates the operation of the isometric clamp.
Here a capturedbead was subjectedto a periodic square
wave of drag force, producedby oscillating the piezo stage
with a periodic triangle wave of displacement.With feedback inactive (left), the bead was deflected in opposite
directionssymmetrically.With activefeedback(right), bead
movementcontinuedup to the clamp set point, after which
the bead position was fixed. When the bead position was
reducedbelow the clamp set point, the systemreturnedto
open-loopmode.Forcescomputedfrom the trap stiffnessin
minimizes local heating,unwantedtrapping of free debris,
and photodamageto the sample.Finally, our system imposescertain limitations. For example,the clamp position
cannotreadily be set to the centerof the trap---becausethe
~E
:1-
2
&
1
"E
5,
Q)
4
E
~
Q
3
(/I
~
~
lm~~ lm~~
Clamp Off
1
-],
i
~
:
2
1
°
i-o.oO'4
L===°
20
~
80
80
100
120
II'~[
]
.200
0
20
~
80
"M
80
100 120
MH MM
J 'YJ\,J ~ ~
0
20
40
80
80
It.oJ
1
°
0
20
40
80
80
100
0,0
200
M~~"C
120
-200
10
8
~ ~
~~ 64
2
2
20
40
Tim8
FIGURE 3
80
(ms)
80
20
~
40
80
80
"'" ~
~
100 120
0
0
20
20
40
~
80
80
Time
0
~
ro
~
80
100
120
100 120
(m8)
Performance of the intensity-modulated position clamp. Pan-
els show the behavior of the system with the feedback system either off
(left) or on (right). In both cases, a freely diffusing bead was first captured
by the trap and then subjectedto a periodic square wave of drag force,
accomplished by moving the stage and specimen with a periodic triangle
wave of displacement (top row). The trap stiffnessremainseither constant,
or is modulatedas required (secondrow). Bead displacement follows a
symmetrical square wave with feedback off, but the maximum excursion is
clipped to the level of the clamp set point (25 nm) when the system is
active (third row). Note the increasedpositional stability (reduction in
displacementnoise)whenthe bead clamped. The measured force produced
by the stage movement is, within error, identical for both the clampedand
unclampedsituations,as expected(bottom row).
Clamp active
: :
indicated
inside
region
:
::: :
I
!: j:
\
\
: :
: !
-150
10
0.5
:g 0.4
.E E
-..c: 0 .3
ci5 Z
CoCo 0 .2
~
~
40
::
-50
-100
0
ro
.=
30
:
~
~ CIa:s~nt
80
-:
""Q. EO:
(/IC:
Q
~
20
( --1~~::~:
50
100 120
\
10
"E
100
Q)
ID
~~J!J1111-M-
4
6
0
0
li~[
100 120
~ J -100f!\Ji~.'J1~\J
10
8
j i
2
~
~
~i02
ft'tI ~ rill! ~
~ 01 JU"U'U.U"urL
~JJJj)jR
-100f
!],
iloO04 lh
~!02
t 01
0.0
200
Clamp On
5
1- :
~~/""""""peTiOd shown)
0
restoring force there is zero, independent of stiffness-and
5
~
,
Trlangle~wavestage displacement
(on
20!
:
!
:
:
iti
!.
'.'i
'.'
l
30
40
Feedback alters stiffness
to clamp position
i:"~:
:!,
t
0.1
:
,
'
0.0
0
10
20
Time
30
40
(s)
FIGURE 4 Stretching a 3888-bp DNA molecule. (Top) The measured
stage displacement in response to one period of a triangle wave voltage
appliedto the piezo stage.Small amountsof curvature,reflectinghysteresis
and nonlinearity, are evident in the piezo output (theseeffects are incorporated in the analysis). (Middle) The displacement record of a bead
tethered to the coverglass by a DNA molecule subjected to stretch. The
feedbackclamp is activatedfor stretchingbeyondabout 60 nm. (Bottom)
The trap stiffnessduring the stretch.Note that it remainsconstantoutside
of the feedbackregion, but rises sharplyto clamp beadposition insidethis
region.
Stretching
Wang et al.
1341
DNA with Optical Tweezers
closed-loop mode were comparableto forces computed
from bead displacementin open-loopmode, as expected.
When used to stretch DNA, the instrument operatesin
two distinct modes,the combinationof which allows optimized measurements
of both weak and strongforcesduring
stretch(Fig. 4). In open-loopmode,before the clamp position is reached,the feedbackis off and the light intensity
remainsconstant;the restoringforce is proportional to the
product of a time-varying bead displacementand a fixed
trap stiffness.Force developsrelatively slowly, which facilitates signal averagingin the noisier, low-force region of
extension.In closed-loopmode,the feedbackis active and
the beadis maintainedat a constantposition; the restoring
force is proportionalto the productof a fixed beaddisplacementanda time-varyingtrap stiffness.Forcedevelopscomparatively quickly, allowing a large dynamic range to be
probed in the high-force region of extension.
Force-extension
relations
Single DNA molecules were stretched repeatedly by fixing
the height of the trap to between 360 and 460 nm and
applying a periodic triangle-wave voltage to the piezo stage
(40-s period). This drive voltage produced stage displacements of 2500-3500 nm (peak to peak). The clamp position
was set to a point located 45-120 nm from the trap center,
so that beads remained within the linear range of both the
detector and trap throughout the stretch. Three data channels
were recorded: position detector output, laser light intensity,
and piezo drive voltage. Analog position signals were lowpass filtered at 1 kHz, digitized at 2 kHz, averaged with a
20-point boxcar window, and stored by computer. Position
and laser signals were converted to Xbeadand Kx, as described in Materials and Methods, using calibration data
obtained with the same bead immediately after each stretch-
50
40
-Z
-
30 ~
Q.
-I
I
I
I
I
I
I
I
I
I
I
I
Q) 20
u
~
0
u.
10
0
200
400
800
600
1000
1200
:
Clamp
:
I
Active
1400
Extension
FIGURE 5 The force-extension relationship for a single DNA molecule in PTC buffer (Table 2). Data (filled circles) were fit to various models of
elasticity, as described in the text. The fit to the modified Marko-Siggia formula (Table 1) is shown (solid line). The feedback system was activated beyond
a DNA extension of ~1300 nm (dotted line and arrow). The contour length of the DNA is ~1314 nm (dashed line). Parameter values and associated errors
for this fit: Lp = 43.3 :t: 0.5 nm, Ko = 1246 :t: 10 pN, Lo = 1314:t: I nm,p = 0.18. Parameter values and associated errors for fits to other models: Odijk:
Lp = 41.2 :t: 0.9 nm, Ko = 1296 :t: 34 pN, Lo = 1316 :t: 1 nm, p = 0.89; Marko-Siggia: Lp = 43.5 :t: 1.0 nm, Lo = 1317 :t: 2 nm, p = 0.12; FJC: Lp
= 3.94 :t: 0.03 nm, Ko = 374 :t: 2 pN, Lo = 1209 :t: 1 nm, p = 0.00.
1342
Biophysical Journal
Volume 72
March 1997
TABLE 1 DNA elasticity models
Model
Fonnula
Marko-Siggia (1995)WLC
=
( ~T\
F
[
Comments
1-
4)
4(1
-
~
xI4f
~ ]
Entropic
4 + 4
theory..Interpolation
10 roM.
Applicable
fonnula
when
F
«
of
exact
exact solution by up to -10% near F
solution
(1995) WLC
Odijk
Smith
[ ~(~ ) ~]
et al. (1995)
FJC
4
-
1
2 FLp
+ K
Lo coth
kBT
2FLp
Marko-Siggia
WLC
(T;;) [
kBT\
-
F -
-
4(1
xI4
i + FIKo)2 -
= coth(a)
1
x
4'+ 4
-
]
F
K:
-
Wormlike
modulus;
ing
chain;
experiment.
with
general,
offset
tion
extension
parity
point
ments
pute
polymerase)
the
and
is
-25
and
in
are
Fig.
Cluzel
al.,
in
an
bead
stage
and
were
then
pN;
has
excess
pN
the
ing
the
a
displace-
used
this
a
simplified
a
persistence
tence
to
shown
its
to
et
results
tions
when
the
that
a
al.,
de-
Fits
to
analytical
polymer
longer
the
Table
may
the
adopt.
Fits
were
The
reflects
method
the
theoretical
have
been
as
a
than
wormlike
derived;
the
by
range
adding
a
stretch-
contributions,
the
in
approach
adopted
by
Odijk
the
or
shown
a
freely jointed chain (FIC), and incorporateentropic and/or
stretching
This
of
theory.
larger
F
2
enthalpic
data
forces
(F
Odijk
fit
range
Fig.
forces
5.
¥
formula
for
F
<
5
pN
p,
obtained
of
for
between
represents
at
were
the
data
only
sets
acquired
15
and
well
at
were
values
to
max
data
>
sets
20
""
stiffness
maximum
from
using
of
the
either
the
quite
0.13);
acquired
pN)
to
comparable.
data
(p
0.19).
""
lower
variant
experimental
Fits
exten(p
excluded
were
force
fit
obtained
This
combined
relative
contributes
pN)
¥
a
large
also
term
applied
(F
Lab-
probability,
of
expressions
the
of
mean
formula
Marko-Siggia:
formula
the
least-
entropic
Data
applicable
pN
and
or
of
results:
purely
deviations
<
max
Parameter
Modified
all
forces.
>
pN,
a
The
This
for
to
nonlinear
following
forces.
value
(1995):
the
in
a
the
is
range
theory,
complete
using
algorithm;
Systematic
for
Marko-Siggia
which
data
1,
experimental
-9
Siggia
experimental
with
the
-0.27.
Data
3.
treat
(WLC)
was
analysis.
two
the
Table
lower
by
occurred
stretching
lon-
stretch
for
the
the
Because
in
to
these
chain
variety
its
interpola-
enthalpic
to
(1995):
for
fit
beyond
The
models
a
broaden
(Levenberg-Marquardt
valid
well
sions.
polymers:
greater
1995)
Instruments),
Marko-Siggia
is
2.
modulus,
wormlike
expressions
Marko-Siggia
Siggia,
in
entropic-enthalpic
configura-
contributions.
length,
Ko' elastic
on
To
intrinsic
of
National
fits
Persis-
greater
elastic
in
summarizes
either
much
various
depending
closely
shown
model
flexible
resistance
enthalpic
correlated
be
compliance
numerous
the
the
and
for
performed
squares
Lo,
uniform,
intrinsic
persistence
1
is
to
may
Ko.
a
Entropic
polymer
polymer's
generally
expressions
backbone
a
length;
theory
View,
tension,
modulus,
that
modified
corresponds
expressions
1996;
DNA
zero
of
direction.
of
the
and
are
elastic
owing
polymer
strain
an
of
under
tendency
same
length
measures
gitudinal
modulus.
the
length,
parameters
the
in
persistence
contrast,
and
the
point
stiffness
length
4,
measures
to
elastic
contour
length,
length
polymer
this
the
its
a
(1995).
were
picture,
contour
assumptions.
(Marko
that
1.
by
(high-force
.
mag-
DNA
(Smith
way
of
by
theory
parameterized
4,
applicability,
appropriate
exceeding
In
«1
com-
example
of
of
we
formula
1996).
elasticity
length;
Note
ranges
term
and
DNA
persistence
theoretical
Odijk
to
stretched
been
60
LoIILo
0.1 pN.
""
applicability,
tion
finding
convert
recently
of
for Ix -
l/a.
4,
underlying
of
for
bead
limited
of
amount.
forces
to
Applicable
contributions.
have
posi-
origin
an
-50
distance);
enthalpic
is
unknown
typically
required
to
(the
by
was
cases,
which
forces
et
DNA
those
form,
with
the
relationships,
5.
some
below
overstretched
velop
data
(end-to-end
hysteresis.
displacement
coordinate
relating
F-x
in
converted
recovered
These
shown
pN,
nitude
the
was
function
Appendix).
single-molecule
which
by
as
load
of
(see
of
x, extension
piezo
stage
position
serves
under
was
for
0.1 pN. Approachesexact
""
forces.
temperature.
for
origin
the
position
F, force;
voltage
corrections
equilibrium
over
latter
drive
coordinate
the
centered
The
stage
chain;
times absolute
appropriate
the
from
jointed
constant
The
displacement
In
FJC, freely
kBT, Boltzmann's
~
Entropic/enthalpic
theory. Modification
of Marko-siggia
fonnula to
incorporate
enthalpic stretching. Has limitations
similar to Marko-
Siggia near F
WLC,
[salt]
from
Entropic/enthalpic
theory. Applicable
to polymers that approximate
FJC. Note that the Kuhn length = 2Lv' and Langevin function
1 + Ko
L(a)
Modified
theory.
when
Differs
regime).
[ (~ )- ~ ]( ~ )
=
x
and higher
Entropic/enthalpic
1/2
=
x
at lower
solution
V4 (kBTK:z!4yJ3.
produced
well
an
at
Marko-
over
the
example
high
parameter
is
maximum
val-
ues similar to those obtainedwith either the Odijk or un-~
Wang et al.
TABLE 2
Stretching DNA with Optical Tweezers
1343
Summary of DNA elastic characteristics
.
;
4 (nm)
Ko (pN)
Lo (nm)
Lo expected
mean :t SE (n)
mean :t SE (n)
mean :t SE (n)
(nm)
10 mM Na+ (NaHPO4 buffer, pH 7.0)
150 mM Na+, 5 mM Mg2+ (NaHPO4 buffer,
47.4:t
1008 :t 38 (10)
1343:t 5 (10)
1328
pH 7.0)
10 mM Na+, 100 JLM spermidine (NaHPO4 buffer,
43.1 :t 1.3 (5)
1205 :t 87 (5)
1348 :t 6 (5)
1328
pH 7.0)
20 mM Tris, 130 mM K+, 4 mM Mg2+ (PfC
38.7 :t 1.0 (8)
1202 :t 83 (5)
1313 :t 2 (5)
1328
buffer, pH 8.0)
20 mM Tris, 130 mM K+, 4 mM Mg2+ (PfC
41.0 :t 0.8 (15)
1277 :t 57 (12)
1352:t 3 (12)
1328
Buffer composition
,
1.0 (14)
,
buffer, pH 8.0)
42.1 :t 2.4 (4)
1010 :t 99 (5)
674:t 5 (5)
661
.
4 values were derived from fits to the Marko-Siggia formula for 0 < F < 5 pN; Ko and Lo values were derived from fits to the Odijk formula for 2 <
F < -25-50 pN. Parameter values returned by either method produced almost identical values for 4 and Lo. The expected value for Lo was computed
from the number of base pairs in the DNA tether, assuming 0.338 nm/bp (Saenger, 1988), plus an allowance of -15 nm to account for linkages at either
end of the DNA tether. Na-phosphate buffers also contained 0.1 mM EDTA. PfC buffer (not all ingredients listed) was used in the assays of Yin et al.
(1995).
modified Marko-Siggia expressions. Fits to partial data sets
that excluded the high-force region (F < 20 pN) produced
some instability in the parameter values. As successively
greater portions of the high-force regime were dropped from
the analysis, Lp increased, whereas both Ko and Lo decreased. It seems likely that this instability develops from
the large difference in stiffness between low- and high-force
regions, and the fact that Ko is mainly determined by elastic
behavior at the very limit of extension.
4. FJC theory: Experimental data were not well fit by this
expression over any force region (p :5 10-6), a finding that
supports conclusions of previous work with double-stranded
DNA molecules (Smith et al., 1992; Bustamante et al.,
1994). The FJC model has, however, been recently applied
with some success to the elasticity of single-stranded DNA
(Smith et al., 1996).
1990; Rau and Parsegian, 1992). Our setup permits direct
visualization of such condensation: when exposed to solutions containing increasing levels of spermidine (in phosphate buffer, pH 7.0, 10 mM Na +), the Brownian motion of
a variable fraction of DNA-tethered beads was suddenly and
dramatically reduced. The remaining fraction of beads continued to diffuse about their tethers. For those tethers that
did not condense, we measured the force-extension relationship. These molecules displayed significantly shorter persistence lengths, even in buffers containing 10 mM spermidine, well below the critical concentration for condensation
(data not shown).
Table 2 summarizes results of fits in a variety of buffer
conditions, including the buffer used for our parallel studies
of transcription by RNA polymerase (PTC buffer; Schafer
et al., 1991). As a control, measurements were carried out in
PTC buffer with two very different sizes of DNA, and the
f .
t
Eft
ec
0
IoniC
.
t
5
th
reng
DNA
on
t " ft
5
I
fits
ness
DNA is a negatively charged polyelectrolyte. Its relatively
long persistence length (-50 nm) compared with base pair
spacing (0.34 nm) may be partly attributable to mutual
electrostatic repulsion of charges along the backbone. Electrostatic screening of such charges is therefore expected to
reduce the persistence length. According to polyelectrolyte
theory, buffers with sufficient concentrations of mono-, di-,
and trivalent ions should neutralize 76%, 88%, and 92% of
charges along DNA, respectively (Manning, 1978). Higher
valence ions, which are more effective in screening charge,
are expected to reduce the persistence length of DNA to a
dist~ce near its "int~nsic:' value. To ~tudy this eff~ct, we
combmed the expreSSIonsm Table 1 WIth our expenmental
returned
similar
values
for
Lp
and
Ko,
but
dramatically
.
dIfferent contour lengths, Lo, as expected. In all cases, fitted
contour lengths were in excellent agreement with the computed lengths of tethers, based on the known number of base
pairs plus a small allowance for the linkages at either end of
the DNA (-15 nm). Persistence length was -47 nm in a
low-sodium buffer (10 mM Na +), dropped to as low as -39
nm in buffers containing either Mg2+ or 100 /LM spermidine3+, but was not further reduced in buffers containing up
to 300 /LM spermidine3+ (data not shown). The elastic
modulus, in contrast, appeared to be independent of buffer
conditions, at least within the accuracyof thesedeterminations.
DISCUSSION
datato extractthe persistencelength andelasticmodulusfor
Persistencelengthscomputedfor 10 mM Na+ buffers were
single DNA molecules in buffers containing Na+, K+,
Mg2+, and spermidine3+.
Spermidine, a trivalent polyamine cation present in a
variety of organisms, is thought to facilitate DNA packing
by directly screening electrostatic forces. In vitro, spermidine binds DNA and induces condensation (Arscott et al.,
significantly longer than those in higher salt buffers. The
persistence length of DNA did not drop below 39 :t 1 nm,
however, even in the presence of multivalent ions expected
to neutralize the vast majority of negative charge along the
DNA backbone, a result which suggests that the intrinsic
persistence length of DNA may be quite close to 40 nm.
1344
Biophysical Journal
Conversely,the elastic modulus was apparentlyinvariant
with ionic conditions,althoughthe larger errors associated
with determiningthis parametermay concealtrends in the
data. If it is true that the elastic modulus remains nearly
constant,then the molecular origin of this parametermay
have little to do with charge screening,but may instead
reflect interactionsintrinsic to the DNA. For an ideal wormlike polymer with uniform, isotropic elastic properties,4
and Ko are intimately coupled, and both are determined
purely by the geometryand Young's modulus.In the simplest possible case, the elastic modulus and persistence
length of a solid rod with circular crosssectionare linearly
related through
K
0
= (~ d2 - )Lp,
..
..
.
whered ISthe diameterof the rod (O~Jk, 1995).~odehng
DNA as such a rod (Hog~ ~d Austin, 1987) Wl~ d = 2
nm ~Saenger,1?88) and pIcking 4 = 4?
(this study)
predictsan elastIcmodulusof ~720 pN, sl~mficantlybelow
the v~ue ~oundhere and elsewhere(SIDlth et al., 1996).
Cho~smgmsteadd = 1.6 nm leadsto Ko = 1100pN, but
~e dis~ce seemsunreason~blysmall. In any event,s.ucha
sImple pIcture cannot explain.the ~ncorrelatedbehavIor~f
and ~o' Clearly, both ela~tIcams~tropyand electrost~tIC
mteractIons mus~ be consIdered m any co~prehensl.ve
m?del of DN~ stIffne~s.For ~x.amp~e,
a ?onumformcyl!ndrical rod WIth a radIal varIatIon m stIffness, becoIDlng
s~fter away ~rom the central ~is, will have a be~ding
stIffn~ss.doIDlnat~dby .~e rela~vely weake~comp.hance
contr1butI~nsat high radiI, andthISp~op:rty will be.dlfectl.y
refle~ted m 4. Ho.we~er,the longitudinal complIance.1S
doIDlnatedby co~tr1butIon~
that do not d~pendon th~ distancefrom .the~lS, and will be reflectedl~ Ko. (Put dl~ferently, bendmgstiffnessd.ependson Y oun~ s ~odul~s mtegrat~dover the cross-sectIonalmomentof mertla,WhIChhas
an r d~pendence,
~hereasthe stretchmodulus~epends.on
Young s modulus mtegratedover the cross sectIon,whi~h
?asan de~ende~ce.)
S~rfaceeffects,suchas electros~tIC
mteractIonsmvolvmg catIons,may thereforeaffect bending
more profoun~l! than stretch.More.accuratemea~ur~ments
o~?NA elastIcity, o~er an ev~n wIder range o~lom~ condltIons, may be requlfed to pm down the relationshIpbetween Ko and 4.
Volume 72
March 1997
been reviewed and compared (Hagerman, 1988). Some
studiesfind that persistencelengthstend toward an asymptotic limit asthe ionic strengthof a buffer containingmonovalent counterionsis increased(Elias and Eden, 1981;Hagerman,1981;Rizzo and Schellman,1981;Maret and Weill,
1983; Borochov and Eisenberg, 1984; Porschke, 1991),
whereasothersreport a more or lesscontinuousdecreasein
persistencelength over the experimentalrangeinvestigated
(Harrington, 1978; Kam et al., 1981; Cairney and Harrington, 1982; Sobel and Harpst, 1991). For those groups
reporting asymptoticbehavior, limiting values of the persistencelength rangefrom 44 to 67 nm (with the exception
of Borochov and Eisenberg,1984,who report 28 nm in a
solution containing 5 M Li+), whereasthe minimal value
found
by the other
groups
from 27 to
om. Studies
havegenerally
found
thatranges
concentrations
of 38
divalent
coun-
terions in the buffer above 0.1 mM do not significantly
reduce persistence lengths below those observed with
monovalentcounterionsabove 10 mM (Hagerman,1988;
but seePorschke,1986). Representativevalues for persistence lengths in Mg2+ solutions include 55 om (basedon
transient electric birefringence; Hagerman,1981), 47 nm
(basedon DNA cyclization and dimerization; Taylor and
Hagerman, 1990), and 44 nm (basedon dichroism; Porschke,1991).The latter value comparesclosely with numbers in Table 2. The effect of polyvalent ions was reported
by two studies that measuredlinear dichroism, where an
asymptoticreductionin persistencelengthfrom 79 nm to 45
om was found after the addition of 10 ILM spermidine
(Baaseet al., 1984),or from 70 om to 30 nm or 20 nm after
the addition of 4 ILM spermineor 10 ILM cobalt hexamine,
respectively(Porschke,1986).The latter valuesfor persistencelength are significantly smallerthan our presentmeasurementsfor spermidine.However, comparisonsamong
bulk methodsare problematicbecauseof systematicuncertaintiesspecificto each:light scatteringandflow dichroism,
for example, rely upon complex data analysis to extract
useful numbers, and corrections for hydrodynamics(excluded volume effects, geometry,self-diffusion, etc.) must
be applied.Electron microscopicmeasurements,
which involve adsorbingthree-dimensionalmoleculesonto a twodimensionalsurface,are subject to still further setsof assumptions (Hagerman, 1988; Bednar et al., 1995).
Consequently,persistencelengthsinferred for similar ionic
conditionsoften differ by factorsof 2 from one studyto the
next. Notwithstanding these difficulties, the present estith
I
"
k
mateof the intrinsic persistencelength (~40 nm) is consisC
.
.
t . h 1.
omparlsons WI ear ler war
ten wit ear ler work.
Determinationsof DNA persistencelength in bulk solution
In contrast with bulk methods, single-moleculetechhavelong beencarried out using a numberof spectroscopic niquesusing optical trapsand associatednanoscaletechnoltechniques,ranging from light scatteringto flow dichroism
ogies(Perkinset al., 1994a,b;Smith et al., 1992,1996;this
to electric or magneticbirefringence.Cyclization studiesof
study) afford an opportunity to make more direct measureDNA in solution can also yield values for persistence mentsof elasticity, free of many of the assumptionsprevilength, as can electronmicroscopicmeasurements
on DNA
ously required. With the new methods,numbers can be
moleculesbound to surfaces.All of these approachesare
obtainedwith greaterspeed,accuracy,and reproducibility.
subjectto a variety of assumptions;previous characteriza- Nevertheless,recoursemust then be made to appropriate
tions of electrostaticcontributions to DNA stiffness have
theoriesto parameterizeelastic behavior in terms of such
~
~
~
I
1346
Biophysical Journal
Volume 72
March 1997
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