Measurement of the DNA Spring Constant Using Optical Tweezers
Charles S. Epstein∗ and Ariana J. Mann
MIT Department of Physics - 8.14
(Dated: May 17, 2012)
An optical trap was used to measure the relationship between restoring force and displacement of
a 1.26 µm polystyrene microsphere attached to a 3.5 kilobase-pair DNA strand. Fitting to a wormlike chain model, the persistence length of the DNA molecule was measured as 53.0 ± 11.6 nm, and
the contour length was measured as 713 ± 36 nm. The Hookean spring constant of DNA, at small
displacements, was found to be 0.162 ± 0.035 pN/µm. These are in satisfactory agreement with
accepted results.
1.
PROBLEM AND RELEVANT THEORY
Deoxyribonucleic acid (DNA) is typically found in a
highly compact “supercoiled” configuration. This is a
result of the higher entropy, and therefore lower Gibbs
free energy, of the compacted state. To stretch DNA thus
requires an input of energy, which implies the presence of
spring-like properties. With an optical trap (“tweezers”),
it is possible to measure the forces exerted by a single
molecule of DNA as it is stretched.
Optical tweezers operate by harnessing the momentum
carried by photons. A laser beam that is highly focused
can create a harmonic “trap” that provides piconewtonscale forces that are relevant to micron-sized objects and
biological molecules. Consider a trap as in Fig. (1).
The arrows on the microsphere (blue) represent “gradient force” vectors, which is caused by light (red) changing its direction as it refracts through the bead [1]. A
“scattering force” balances this, which is a pressure on
the bead resulting from reflections (not shown in the image). These forces combine to produce a stable equilibrium with a near-harmonic region at the center [1].
Id: 51.opticaltrap.tex,v 1.11 2012/02/06 23:45:01 spatrick Exp
2
For objects much larger than the wavelength of the
light, the force can be described with ray optics; for objects that are much smaller, the forces can be quantified
with Rayleigh scattering [1]. However, the 1.26 µm microspheres manipulated in this experiment with 975 nm
laser light are not precisely described with either theory.
The principles of the motion can be generally understood
by these models, while the precise bead dynamics are elucidated via calibrations.
1.1.
Models to Describe DNA Spring Properties
The entropic spring-like forces of DNA can be understood by a worm-like chain (WLC) model. This approximates the DNA strand as a continuously bendable thin
chain. To understand this model, it is useful to first approach a discrete approximation to it, the freely-jointed
chain (FJC). Following Storm and Nelson [2], we consider
a chain with N segments of length b, with orientation unit
vectors {t̂i }, subject to a force f~ = f ẑ (Fig. 2). We find
the energy E of the system to be
They can
ic spheres
nside bio-
N
X
E F JC ({t̂i })
fb
=−
t̂i · ẑ
kB T
k
T
i=1 B
overed by
at the raed entirely
would ac2
. When
diction, he
ccelerated
attracted
f the samrap by usbacterium
ped in the
use in cell
ly in both
abs world-
with kB Boltzmann’s constant and T the temperature
[2]. A relation between the extension z and applied force
f can be derived from Eq. (1), which takes the form [2]
xplore the
heres held
nion cells.
he scale of
hydrodycrospheres
trap prophysics of
DzE
L
FIG. 1:
1. A
ray diagram
showing
the gradient
force stabiFIG.
Diagram
showing
the how
gradient
forces applied
to a
lizes themicrosphere.
trap laterally In (a) the off-center bead feels a force
trapped
toward the center. In (b), the bead is in equilibrium with an
upward scattering force (not pictured). From [1].
In order to understand how the equilibrium is stable,
it will help to consider how the gradient force responds to
displacement of a bead from the center. As seen in Figure
the red region represents the “waist” of the laser at
∗1,
Electronic address: [email protected]
its focus point, with the laser passing upward through
the sample chamber. The blue ball is the bead, and the
dark red arrows (1) and (2) represent light rays whose
thicknesses correspond to their intensities (note that the
beam is brightest at its center). In case (a), with the
particle slightly to the left of center, the two rays refract
= coth
fb
kB T
−
kB T
fb
(1)
(2)
where L is the total length of the chain. This model
is a crude simplification of the DNA strand and is not
generally used, as the force-extension relation does not
accurately describe physical results at higher forces [2].
Making the leap from discrete sections to a continuous
strand provides the more widely accepted WLC model.
In this model, the resistance to bending is also addressed.
As noted in Fig. (3), the position along the chain is
denoted as ~r(s), and local curvature and tangent vectors
w(s)
~
and ~t(s) are defined, according to [2], as
C. STORM AND P. C. NELSON
PHYSICAL REVIEW E2 67, 051906
PHYSICAL REVIEW E 67, 051906 &2003'
C. STORM AND P. C. NELSON
FIG. 1. The freely jointed chain consists of identical segments
of length b, joined together by free hinges. The configuration is
fully described by the collection of orientation vectors " t̂ i # . " - i #
FIG.
1. 2:The
freely
jointed chain
chain
consists
of
identical
segments
FIG.
The
freely-jointed
consisting
Nthe
disFIG. 2. A wormlike chain is a continuum elastic medium, whose
denotes
the angle between
t̂ i model,
and the fixed
directionof
ẑ of
applied
3: The worm-like
chain
a continuous
creteb,
sections
of together
length
From
[2]. hinges. The configuration FIG.
is described
in model,
terms of consisting
the positionofvector
r! as a
length
joined
free
is configuration
stretching
force. b. by
strand
withoftotal
length
function
contour
lengthLs.tot = lc . From [2].
of
fully described by the collection of orientation vectors " t̂ i # . " - i #
extension relation of the model. The fit value of b can then
FIG. 2. A
chain
B. wormlike
The wormlike
chainis a continuum elastic medium
denotes the angle
between
and
the fixed
direction
ofitsthe
applied
depend
both ont̂ ithe
molecule
under
study andẑ on
external
light isabove,
then
directed into
a quadrant
configuration
is double-stranded
described
in terms
of thephomentioned
DNA
&dsDNA'
isposition vecto
stretching force.conditions such as salt concentration, as those conditions af- (CD).AsThe
~t(s)
todetector
(QPD).
The
QPD
consists
ofis not
foursurprising
photodid
d~
r
(s)
far
from
being
a
freely
jointed
chain.
Thus
it
fect the
intramolecular
interactions.
function
of
contour
length
s.
~t(s) =
, w(s)
~
=
.
(3)
thatand
while
FJCvoltages
model can
the observed
linear
To formulate ds
the FJC, we describe
hasthetwo
as reproduce
output: the
sum of the
two
ds a molecular conforma- odes,
force-extension
relation
dsDNA
at low
force,
tion byof
associating
with each
segment
a unit orientation
extension relation
the model.
The
fit value
of b canvecthen
upper
voltages minus
theofsum
of the
twostretching
lower voltages,
B.
wormlike
observed
saturation
highThe
force,
still
failschain
at inThe
inextensibility
of the
is study
enforced
the
tor the
t̂ i , pointing
in chain
the
direction
of and
thebyith
segment,
as andand
thethesum
of the
two leftat minus
the
twoit right.
This
depend
both
on
molecule
under
on
itsconexternal
termediate
values
of
f.
Another
indication
that
the
model
is it
~
dition that
|t(s)| in= Fig.
1 at1. all
points
alongof the
chain [2].
!f provides a measurement of the deflection of the light as
sketched
In
the
presence
an
external
force
mentioned
double-stranded
conditions
such as
salt
concentration,
as
those
conditions af- physicallyAs
inappropriate
is thatabove,
the best-fit
value of the Kuhn DNA &dsD
Noting that
the
energy
of
an
elastic
rod
is
proportional
through the sample, which is proportional to bead
along the ẑ direction, we can define an energy functional for passes
segment
isbeing
b+100anm,
completely
different
from
the it is not su
far length
from
freely
jointed
chain.
Thus
fect the
intramolecular
to the
square
of theinteractions.
local curvature, and retaining the
displacement
within
the trap.
the chain
physical contour length per basepair of 0.34 nm.
that
while
the
FJC we
model
can reproduce
the observe
Toright-hand
formulateside
the ofFJC,
a molecular
Eq. we
(1),describe
the energy
of the chainconformacan
TheTosample
blue
improvewas
uponvisually
the FJC, imaged
must using
accounta for
the lamp.
fact
N
FJC
t̂
f
b
E
!
$
"
#
be
expressed
as
[2]
i
force-extension
relation
of
dsDNA
at
low
Thethat
light
through
thebending.
sampleInwith
tion by associating with each segment
a unit
orientation &1'
vecthe passes
monomers
do resist
fact,orientation
the very greatop- stretching
!" %
t̂ i •ẑ.
k BT
i!1 k BT
posite
the
laser,
through
a
focusing
lens,
into still it fail
stiffness
double-stranded
can be turned
tothen
our
adandofthe
observed DNA
saturation
at and
high
force,
tor t̂ i , pointing in the direction of the ith !
segment, aasCCD
vantage,
as
it
implies
that
successive
monomers
are
concamera,
which
allows
the
sample
to
be
visualized
Z
termediate
values of f. Another indication that the m
dt̂(s) 2 off an external force on
! computer
[t̂(s)] absence
l presence
to point
in nearly the same direction. Thus we can
sketchedE W
inLCFig.
In lcthe
screen.
external
−
In the1.
of pan
force, allt̂(s)
configurations
have f astrained
=
·
ẑ
ds
physically
isinbody,
that
theconfigurabest-fit
ds self-avoidance'
treat stage
the polymer
asinappropriate
a was
continuum
elastic
its
2&neglecting
T
position
controlled
a rough
mannervalue of th
equal energy
the chain dis-forThe
BT
0 and
along the ẑ kdirection,
we
can
define ankBenergy
functional
!
(s)
as
a
function
of
tion
described
by
the
position
r
segmentmicrometers,
length is b+100
nm, completely
plays the characteristics of a random walk. To pull the
and precisely
by piezotheac- different f
(4)ends by hand-turned
the chain
relaxed-state
contour
length
s
&see
Fig.
2'.
Continuing
to
of such a chain away from each other a force has to be tuators.physical
The piezo
positionlength
and laser
wereofconcontour
per power
basepair
0.34 nm.
treat the chain as inextensible gives the wormlike chain !4,5$.
as extending the chain reduces its conformational trolled within a MATLAB interface, which also handled
where lc isapplied,
the FJC
total
“contour” length
of the chain, and lp
improve
upon vectors
the FJC,
wew! , must
account for
N
The
localTo
tangent
and curvature
( !t and
respecentropy.
The
entropic
behavior can be sum- data
t̂resulting
felastic
bwhich
E !length
collection.
Calibrated
strain gauges
yielded
a pre$ of the
"
#
is the persistence
chain,
is the length
i
that
the
monomers
do
resist
bending.
In
fact, the ver
tively'
are
given
by
marized in the force-extension
relation
!3$
!"
t̂ i •ẑ. chain seg- &1'
cise measurement of the x-y stage position. The maxiscale at which directional
correlations
between
k BT
i!1 k BT
stiffnesslaser
of double-stranded
DNA can be turned to
mum attainable
power
! & s ' was 100
ments decay. Since this is not soluble exactly, interpodr
d !tmW,
& s ' however data
z
fb
k BT
!
!
s
!
s
!
,
w
.
&4'monomers
t
'
'
vantage,
as
it
implies
that
successive
a
&
&
was
acquired
at
5
and
10
mW
to
avoid
the
application
of
lations of numerical approximations
an ac- &2'
,
!coth have
" yielded
ds
ds
L
k
T
f
b
excessive
forces
to
the
DNA
strand.
tot
B
cepted functional form [3]
strained to point in nearly the same direction. Thus
%
! " # $
In the absence of an external force, all configurations have We temporarily assume that the chain is inextensible, extreat the polymer as a continuum elastic body, its co
equal energy the
andwell-known
&neglecting
self-avoidance'
chain
dis- pressed
function. In #thethe
limit
of low
locally by the condition that % !t (s) % !1 everywhere.
" Langevin
−2
as a function
by generalizing
the position
r! (s)
stretching
all apolymer
to the
Hooke-law
get andescribed
energy
Eq. &1',
we note
plays the characteristics
random
walk.
pull
the ends Totion
z models
z To
1of
1 reduce
kB Tforce,
2.1. functional
Sample Preparation
+
. spring constant
(5)
f=
we define −
the effective
behavior
f !k sp( z )1; −
a thin, homogeneous
rod thelength
elastic energy
density
is 2'. Contin
relaxed-state
contour
s &see
Fig.
lp
4from each
lc
4 alc force has to be that for
of such a chain
away
other
L tot , or
by * !k sp
proportional to the square of the local curvature. Adding the
treat
the
chain
as
inextensible
gives
the
wormlike
chai
tethers term
werefrom
prepared
applied, as extending the chain reduces its conformationalDNA
external-force
Eq. &1'through
yields an involved series
ThisThe
formula
works entropic
well for f <
5 pN and
lp lccan
[3]. be
At sum!,
of stagesThe
(as local
in [3]) tangent
that attached
1.26 µm streptavidinand curvature
vectors ( !t and w
entropy.
resulting
behavior
z elastic
f
2
2spring constant
L tot
low z, it reduces to an effective →
Hookean
coated polystyrene
(kb)
A d t̂ &to
s ' 3.5 kilobase-pair
f
E WLC! t̂ & s '$ microspheres
#O
f
.
&3'
'
&
tively' are! given
marized
in the force-extensionL totrelation
!3$
*
t̂ & s ' •ẑ . &5'
ds by
"
of
! "
&
'( (
)
DNA, the other
was attached
to a glass
k BT end 0of which
2 ds
k BT
coverslip. The DNA strand was prepared first by perd !t & s '
Expanding Eq. &2' gives the effective spring constant for the forming
' onofa the
&asmeasure
Equation
&5' shows that
parameter
Adr
is!(PCR)
bend
polymerase
chain
reaction
sequence
FJC
z
f
b
T
k
!
!
3k
T
, wof
.
'!
& s '!
& s PCR
FJC as * !3k BT/b. The Bfact that B
the effective spring con- within
stiffness
the chain. plasmid.
A ist also
the
persistence
of the
the of
M13mp18
The
can
κW LC =
.
(6)
,
!coth
"
dsprocesslength
ds
stant isL proportional
tokthe
temperature
illustrates&2'
chain, the characteristic
lengthsequence.
scale associated
with
the
de- is
2lT
p labsolute
c
significantly
amplify
a
DNA
First,
the
DNA
f
b
B
that the tot
elasticity in this model
is purely entropic in nature.
cay of
zerostrands;
stretchingthe
force:
heated
totangent-tangent
denature thecorrelations
DNA intoattwo
temAt high stretching force, Eq. &2' gives ( z/L tot) →1; the
We
temporarily
assume
that
the
chain
is inextensib
perature
is
then
lowered
so
that
primers
may
anneal
to
extension saturates when all the links of the chain are aligned
t̂ & 0 ' • t̂ & s ' ) WLC,e " % s % /A .
&6'
(
the endspressed
of the sequence
the well-known
In the links
limit
of low
locally by
bycomplimentary
the conditionbase-pairing.
that % !t (s) % !1 every
by theLangevin
external force.function.
In reality, individual
are slightly
2. EXPERIMENTAL
SKETCH
AND
SALIENT
The
temperature
is
then
raised
slightly
to
induce
DNA Eq. &1', w
extensible;
we
will
modify
the
model
to
introduce
this
effect
The
force-extension
relation
for
the
WLC
was
obtained
stretching force, all polymer
models reduce to the Hooke-law
To get an energy functional generalizing
DETAILS
polymerase
toinadd
to extend
in Sec. II C.
numerically
Ref.complimentary
!6$; subsequently nucleotides
a high-precision
interbehavior f !k sp( z ) ; we define the effective spring constant
that forThis
a thin,
homogeneous
rod30the
elastic
the sequence.
process
was repeated
times
us- energy de
, or of the optical trap used in this experiment 051906-2
by * !kThe
proportional
to thecycler
square
of the local
curvature. Add
ing an automated
thermal
(a four-hour
process),
layout
spL tot
providing
an amplification
factor
approximately
can be seen in Fig. (4). The 975 nm near-infrared laser
external-force
term
fromof Eq.
&1' yields 230 .
The primers for one end of the DNA strand were funcis collimated and directed through a 100x objective lens
tionalized with WLC
biotin, which hasL a high affinity for2 the
(OBJ), through thez sample,f and into
a
condenser
lens
2
! " # $
! "
L tot
→ #O & f ' .
*
&3'
Expanding Eq. &2' gives the effective spring constant for the
FJC as * FJC!3k BT/b. The fact that the effective spring con-
E
! t̂ & s '$
!
k BT
&
0
tot
ds
'( (
A d t̂ & s '
f
t̂ & s ' •ẑ
"
2 ds
k BT
)
Equation &5' shows that parameter A is a measure of t
stiffness of the chain. A is also the persistence length
3
remove their wax-like coating. This allowed the antidigoxigenin antibodies to bind to the surface. The glass
slides were not etched, thus ensuring that only one side
of the sample would contain DNA tethers.
DNA-microsphere complexes were then prepared by
making a 1:1 mixture of 20 picomolar DNA and 1%
(weight) microspheres, and incubating 4 hours at 4◦ C.
The solution was stored at -20◦ C prior to use.
Flow cells were then constructed by placing two strips
of double-stick tape side-by-side on a glass slide, which
were then covered with an etched coverslip (Fig. 5).
DNA tethers were then attached to the flow cells. First, a
1:5 dilution of 20 mg/mL anti-digoxigenin in phosphatebuffered saline (PBS) solution was made. This was further diluted 1:10 in PBT, a solution containing PBS,
bovine serum albumin, and Triton-X (a surfactant). A
25 µL volume of the anti-digoxigenin solution was flowed
into the cell using a vacuum, and incubated for 40 minutes at room temperature. Following this, 200 µL of a
1 mg/mL casein (milk protein) in PBT solution, cleaned
through a syringe filter, was flowed through the cell in
order to block the binding of other molecules to the coverslip. The sample was then incubated 20 minutes at
room temperature. Next, 25 µL of bead-DNA complexes
were flowed into the cell, and incubated for 20 minutes
at room temperature. Finally, 800 µL of the casein solution was flowed through the cell to wash out unbound
beads. The sample was then sealed with vacuum grease
and immediately analyzed with the optical trap. Rapid
degradation of the samples necessitated the construction
of new ones for each day of data acquisition.
FIG. 4: Optical configuration of the trap, based on figure
from [4].
2.2.
FIG. 5: Flow channel between coverslip, slide, and doublestick tape. Based on figure from [4].
streptavidin molecules on the microspheres. The primers
for the opposite end were functionalized with digoxigenin,
which binds to anti-digoxigenin antibodies later attached
to the coverslip. After the PCR reaction, agarose gel
electrophoresis confirmed that the product was 3.5kb in
length, and then the DNA was cleaned using a Qiagen
QiaQuick kit, which removed the DNA polymerase and
unused nucleotides from the solution.
The glass coverslips were etched in a 1:1 solution of
potassium hydroxide (KOH) and ethanol in order to
Trap Calibrations
The action of the optical trap on the 1.26 µm
polystyrene beads was characterized with automated
software written by Dr. S. Wasserman. The Brownian motion of a bead in the trap was analyzed near the
coverslip because of the eventual measurement at that
location. A value of the trap stiffness α was found by
analyzing the power spectral distribution (PSD) of the
frequencies present in the Brownian oscillations.
Under the theory of Brownian motion, we expect that
a particle in thermal equilibrium at temperature T will
undergo random motion under what can be considered a
random force F (t). Since the force is random, we expect
it to have a spectrum of white noise [1]. For a bead
at low Reynolds number, as in this experiment, viscous
drag forces always dominate over the bead’s inertia; this
means that we can express the force on the bead as a
function of time as
β ẋ + αx = F (t)
(7)
as in [1], where α is the spring constant of the trap,
β = 3πηd is the hydrodynamic drag coefficient, η is the
medium’s viscosity, and d is the diameter of the bead. A
4
QPD Response vs Strain Gauge Stage Position
PSD function can be defined using the Wiener-Khinchin
theorem and the Fourier transform of the time-averaged
autocorrelation function, following [1], leading to
kB T
2
π β(f 2 +
f02 )
(8)
in which f0 = α/2πβ. Thus, by examining the PSD of
a bead’s Brownian motion, the spring constant α of the
trap can be inferred. This yielded α = 3.1 ± 0.2 × 10−6
N/m at 5 mW and 3.2 ± 0.2 × 10−6 N/m at 10 mW. By
using the equipartition theorem, which holds that the energy in a harmonic oscillator 12 αhx2 i = 12 kB T , the QPD
responsivity R (a conversion between QPD voltage and
bead displacement) was found to be R = 9.1 ± 0.2 × 105
V/m at 5 mW, and 1.6 ± 0.2 × 106 V/m at 10 mW. The
conversion between stage position and strain gauge voltage was provided by 20.309 staff as 2.22 V/µm.
2.3.
DNA Tether Measurements
DNA tethers were identified by their highly-localized
Brownian motion. Once a bead was trapped, the stage
was oscillated at a low frequency (to minimize viscous
drag forces) and QPD displacement data was recorded.
This provided a measurement of bead displacement
within the trap (and thereby force) as a function of the
length to which the DNA was stretched. The measurement was optimized by adjusting the x-y stage position
to center the bead within the trap, performed by visually observing the bead, as off-center beads oscillated in a
kinked path. The stage height was then adjusted in order
to position the bead as close to the coverslip as possible
by observing the QPD response: as the bead approached
the coverslip, the maximum stretching distance increased
until the bead made contact, which then caused the data
to lose its characteristic shape. The data was acquired
at a height just before this point.
QPD Response Voltage [V]
Sxx (f ) =
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
4
4.5
5
5.5
6
6.5
Strain Gauge Stage Position [V]
FIG. 6: Raw data: QPD voltage (measure of bead deflection)
versus strain gauge voltage (measure of stage position). The
center of the graph corresponds to zero extension. The curve
shows the low-spring constant regime of the DNA (flatter center portion), the stretching as it approaches the strand length
(steeper slope), and finally the bead being pulled out of the
trap (maximum/minimum).
DNA Restoring Force versus Tether Extension
0.4
0.35
Restoring Force [pN]
s
0.3
0.3
0.25
0.2
0.15
0.1
0.05
0
0
100
200
300
400
500
600
Tether Extension [nm]
3.
DATA PRESENTATION AND ERROR
ANALYSIS
FIG. 7: DNA force-extension curve. Persistence Length:
42.9 ± 0.7 nm, Contour Length: 812 ± 3 nm, χ2r = 1.0.
Five tethers were analyzed, at laser powers of 5 mW
and 10 mW. The bead position was oscillated with frequency 0.55 Hz. QPD voltage as a function of strain
gauge voltage was recorded; an example plot of this data
can be seen in Fig. (6). The strain gauge voltage was
then converted to stage position, and the QPD voltage was converted to bead displacement within the trap.
From these, the DNA extension z was calculated using
the relation [3]
z=
q
h2 + (xstage − xbead )2 − r
(9)
which depends on the bead height h and the measured
quantities xstage and xbead (Fig. 8). The height h was
taken to be h = r since the beads were positioned directly
adjacent to the coverslip. The restoring force was calculated by first converting QPD voltage to bead displacement with R, and then converting bead displacement to
trapping force with α. A plot of restoring force versus
tether extension for one tether is visible in Fig. (7). The
data has been binned into equally-spaced values of the
tether extension; error bars result from the standard error within each bin as well as from the calibrations.
5
FIG. 8: Diagram showing DNA extension z in relation to
bead position. Here, xstage − xbead of Eq. (9) is represented
by the quantity d.
Across the five tethers, the value of the DNA persistence length lp was measured as 53.0 ± 11.6 nm, and the
contour length lc was measured as 713 ± 36 nm. With
the regime of small-displacement, the Hookean spring
constant of DNA, following Eq. (6), is found to be
0.162 ± 0.035 pN/µm. The uncertainties reflect the standard error of multiple measurements as well as the uncertainties from the fits.
4.
DISCUSSION
Accepted values for lp are between 40-50 nm [3], and
for 3.5kb DNA, the accepted value of lc is approximately
1180 nm [3]. Our agreement for the persistence length
is quite satisfactory. The expected value for the smalldisplacement Hookean spring constant is 0.104 pN/µm;
our result deviates by 1.6σ. Our measured value of the
contour length is low by 13σ, however, the order of magnitude is as expected. A possible explanation is the extreme sensitivity of the calibrations to the height of the
microsphere; however, without precise h measurements
it is difficult to quantify this. It is difficult to measure h
precisely because there is no piezoelectric control of the
[1] Junior Lab Staff, Optical Trapping Lab Guide (2012).
[2] C. Storm and P. Nelson, Physical Review E 67 (2003).
[3] D. Appleyard, K. Vandermeulen, H. Lee, and M. Lang,
Am. J. Phys. 75 (2007).
[4] 20.309 Staff, Optical Trapping Lab Guide (2006).
[5] M. D. Wang, H. Yin, R. Landick, et al., Biophysical Journal 72, 1335 (1997).
Acknowledgments
The author gratefully acknowledges Ariana Mann,
Gustaf Downs, and Devin Cela for their equal part in per-
stage height nor a way to accurately measure vertical
bead deflection.
A further systematic uncertainty is the possible damage of the DNA strand due to multiple oscillations of the
bead position. Care was taken to acquire data as soon as
possible once the stage was set to oscillate. This could
be addressed in future iterations of this experiment by
writing a stage oscillation protocol that would find the
center of DNA attachment and record QPD data with
the fewest possible number of oscillations. A possible
method for this is to sweep the bead in the y direction,
identify the center, and then perform one extended sweep
in the x direction.
Another cause of concern was the presence of small
foreign particles in the flow cells, which were occasionally
drawn into the trap during measurement. The data was
not seen to fluctuate significantly when this occurred,
however, in the future, more meticulous filtration of the
solutions may be desirable.
5.
CONCLUSIONS
The primarily entropic spring-like properties of DNA
were observed by manipulation of a single molecule in an
optical trap. With a fit to a worm-like chain model, the
persistence and contour lengths of a 3.5kb DNA strand
were measured to be 53.0 ± 11.6 nm and 713 ± 36 nm, respectively. The former is in agreement with the accepted
range of 40-50 nm, while the latter, despite being correct
in order of magnitude, deviates by 13σ. The Hookean
spring constant of DNA, at small extension distances,
was found to be 0.162 ± 0.035 pN/µm, a 1.6σ deviation
from the expected value. Future extensions of the experiment could test different stretching regimes of DNA or
other force-extension models, such as those that include
corrections for aqueous ion interactions or enthalpic properties [5].
forming this experiment, which included many full days
of preparation. The author would also like very much to
thank Dr. Steve Wasserman and Dr. Steven Nagle who
were invaluable at every stage of this experiment.