Measurement of the DNA Spring Constant Using Optical Tweezers

Charles S. Epstein∗ and Ariana J. MannMIT Department of Physics - 8.14

(Dated: May 17, 2012)

An optical trap was used to measure the relationship between restoring force and displacement ofa 1.26 µm polystyrene microsphere attached to a 3.5 kilobase-pair DNA strand. Fitting to a worm-like chain model, the persistence length of the DNA molecule was measured as 53.0± 11.6 nm, andthe contour length was measured as 713 ± 36 nm. The Hookean spring constant of DNA, at smalldisplacements, was found to be 0.162 ± 0.035 pN/µm. These are in satisfactory agreement withaccepted results.

1. PROBLEM AND RELEVANT THEORY

Deoxyribonucleic acid (DNA) is typically found in ahighly compact “supercoiled” configuration. This is aresult of the higher entropy, and therefore lower Gibbsfree energy, of the compacted state. To stretch DNA thusrequires an input of energy, which implies the presence ofspring-like properties. With an optical trap (“tweezers”),it is possible to measure the forces exerted by a singlemolecule of DNA as it is stretched.

Optical tweezers operate by harnessing the momentumcarried by photons. A laser beam that is highly focusedcan create a harmonic “trap” that provides piconewton-scale forces that are relevant to micron-sized objects andbiological molecules. Consider a trap as in Fig. (1).The arrows on the microsphere (blue) represent “gradi-ent force” vectors, which is caused by light (red) chang-ing its direction as it refracts through the bead [1]. A“scattering force” balances this, which is a pressure onthe bead resulting from reflections (not shown in the im-age). These forces combine to produce a stable equilib-rium 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

temperature to a few micro-Kelvin and below. They canalso be used to push or trap microscopic dielectric spheres— or even entire, living, cellular organisms, inside bio-logical media.

The method of optical trapping was discovered byArthur Ashkin in 1970 [5] [6]. He calculated that the ra-diation pressure from a high power laser, focused entirelyonto a micron-sized bead (or “microsphere”), would ac-celerate the bead forward at nearly 106 m/s2. Whenhe performed the experiment to test this prediction, hefound that while the target bead was indeed accelerateddownstream, other beads in the solution were attractedlaterally into the beam-path from other parts of the sam-ple. He then created the first working optical trap by us-ing two opposing laser beams. At one point a bacteriumthat had contaminated a sample became trapped in thebeam, thus instigating the trap’s revolutionary use in cellbiology. Today optical traps are used extensively in bothatom-trapping experiments and in biophysics labs world-wide.

In this laboratory experiment, you will explore theuse of optical forces to trap dielectric microspheres heldwithin a thin layer of water and vesicles in onion cells.The typical mechanical forces involved are on the scale ofpiconewtons (10�12 N). Relative to this scale, hydrody-namical forces (drag and di↵usion) on the microspheresand vesicles are substantial. Thus, the optical trap pro-vides an excellent opportunity to study the physics ofBrownian motion, which you will use to obtain a quan-titative measurement of Boltzmann’s constant. In theprocess, you will calibrate the dependence of trap sti↵-ness (force/distance) on laser supply current. Biologi-cal motors, which are vital to intracellular transport andbacterial locomotion, also act with forces on this scale.You may thus employ the optical trap to quantify thespeed and force of a molecular motor moving a vesiclealong an actin fiber in an onion cell.

I.1. The Physics of Optical Trapping

The following material in this subsection is takennearly verbatim from UC Berkeley’s Junior Lab guideon their optical trap experiment [2].

The most straightforward mechanism to understandthe physics of trapping is to consider the change in mo-mentum of light that is scattered and refracted by thedialectic material, in our case a silica glass bead. Anychange in the direction of light imparts momentum tothe bead. This mechanism holds for objects much largerin diameter than the wavelength of the laser. A ray-tracing argument implies that the scattered light createsa scattering force in the direction of light propagation,while the refracted light creates an opposing gradientforce. When the bead is in the center of the trap, theseforces cancel. When a bead moves slightly away from thecenter, a net force is applied towards the center, makingthis a stable equilibrium.

FIG. 1. A ray diagram showing how the gradient force stabi-lizes the trap laterally

In order to understand how the equilibrium is stable,it will help to consider how the gradient force responds todisplacement of a bead from the center. As seen in Figure1, the red region represents the “waist” of the laser atits focus point, with the laser passing upward throughthe sample chamber. The blue ball is the bead, and thedark red arrows (1) and (2) represent light rays whosethicknesses correspond to their intensities (note that thebeam is brightest at its center). In case (a), with theparticle slightly to the left of center, the two rays refractthrough the particle and bend inwards. The reactionaryforce vectors, F1 and F2, of each ray on the bead areshown as black arrows. Because ray (2) is more intense(and thus carries more momentum) than ray (1), the netforce on the bead is to the right. Thus, a perturbationto the left causes a rightward-directed force back towardsthe trap’s center.

In case (b) the particle is centered laterally in the beamand will not be pushed left or right. The net gradientforce is downward, which is balanced by an upward scat-tering force (not shown) due to reflection of some of thelight.

To better understand how the scattering and gra-dient forces and the trap’s stability vary with beaddisplacement both vertically and horizontally, trythis Java applet (http://glass.phys.uniroma1.it/dileonardo/Applet.php?applet=TrapForcesApplet)from the DiLeonardo lab [3] in Italy. The model usedfor this applet shows the importance of a high numericalaperture lens, as the extremal rays illustrated contributedisproportionately to the change in gradient forcevertically. (Note that you must adjust the numericalaperture at the bottom of the applet in order to obtaina stable trap.) By moving the bead around and lookingat the net force vector, you can get a pretty good feelfor how the restoring force varies as a bead is displacedhorizontally or vertically from the trap’s center. Noteparticularly how the trap is less sti↵ as the bead isdisplaced above the trap’s center. Remember this whenyou trap your first bead and try moving the bead with

FIG. 1: Diagram showing the gradient forces applied to atrapped microsphere. In (a) the off-center bead feels a forcetoward the center. In (b), the bead is in equilibrium with anupward scattering force (not pictured). From [1].

∗Electronic address: cepstein@mit.edu

For objects much larger than the wavelength of thelight, the force can be described with ray optics; for ob-jects that are much smaller, the forces can be quantifiedwith Rayleigh scattering [1]. However, the 1.26 µm mi-crospheres manipulated in this experiment with 975 nmlaser light are not precisely described with either theory.The principles of the motion can be generally understoodby these models, while the precise bead dynamics are elu-cidated via calibrations.

1.1. Models to Describe DNA Spring Properties

The entropic spring-like forces of DNA can be under-stood by a worm-like chain (WLC) model. This approx-imates the DNA strand as a continuously bendable thinchain. To understand this model, it is useful to first ap-proach a discrete approximation to it, the freely-jointedchain (FJC). Following Storm and Nelson [2], we considera chain withN segments of length b, with orientation unit

vectors {ti}, subject to a force ~f = fz (Fig. 2). We findthe energy E of the system to be

EFJC({ti})kBT

= −N∑i=1

fb

kBTti · z (1)

with kB Boltzmann’s constant and T the temperature[2]. A relation between the extension z and applied forcef can be derived from Eq. (1), which takes the form [2]

⟨ zL

⟩= coth

(fb

kBT

)− kBT

fb(2)

where L is the total length of the chain. This modelis a crude simplification of the DNA strand and is notgenerally used, as the force-extension relation does notaccurately describe physical results at higher forces [2].Making the leap from discrete sections to a continuousstrand 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 isdenoted as ~r(s), and local curvature and tangent vectors~w(s) and ~t(s) are defined, according to [2], as

2

extension relation of the model. The fit value of b can thendepend both on the molecule under study and on its externalconditions such as salt concentration, as those conditions af-fect the intramolecular interactions.To formulate the FJC, we describe a molecular conforma-

tion by associating with each segment a unit orientation vec-tor t i , pointing in the direction of the ith segment, assketched in Fig. 1. In the presence of an external force f!along the z direction, we can define an energy functional forthe chain

E FJC!" t i#$

kBT!"%

i!1

N f bkBT

t i• z . &1'

In the absence of an external force, all configurations haveequal energy and &neglecting self-avoidance' the chain dis-plays the characteristics of a random walk. To pull the endsof such a chain away from each other a force has to beapplied, as extending the chain reduces its conformationalentropy. The resulting entropic elastic behavior can be sum-marized in the force-extension relation !3$

! zL tot" !coth# f b

kBT$"

kBTf b , &2'

the well-known Langevin function. In the limit of lowstretching force, all polymer models reduce to the Hooke-lawbehavior f!ksp(z); we define the effective spring constantby *!kspL tot , or

! zL tot"→

f*

#O& f 2'. &3'

Expanding Eq. &2' gives the effective spring constant for theFJC as *FJC!3kBT/b . The fact that the effective spring con-stant is proportional to the absolute temperature illustratesthat the elasticity in this model is purely entropic in nature.At high stretching force, Eq. &2' gives (z/L tot)→1; the

extension saturates when all the links of the chain are alignedby the external force. In reality, individual links are slightlyextensible; we will modify the model to introduce this effectin Sec. II C.

B. The wormlike chain

As mentioned above, double-stranded DNA &dsDNA' isfar from being a freely jointed chain. Thus it is not surprisingthat while the FJC model can reproduce the observed linearforce-extension relation of dsDNA at low stretching force,and the observed saturation at high force, still it fails at in-termediate values of f. Another indication that the model isphysically inappropriate is that the best-fit value of the Kuhnsegment length is b+100 nm, completely different from thephysical contour length per basepair of 0.34 nm.To improve upon the FJC, we must account for the fact

that the monomers do resist bending. In fact, the very greatstiffness of double-stranded DNA can be turned to our ad-vantage, as it implies that successive monomers are con-strained to point in nearly the same direction. Thus we cantreat the polymer as a continuum elastic body, its configura-tion described by the position r!(s) as a function of therelaxed-state contour length s &see Fig. 2'. Continuing totreat the chain as inextensible gives the wormlike chain !4,5$.The local tangent and curvature vectors ( t! and w! , respec-tively' are given by

t!&s '!dr!&s 'ds , w! &s '!

d t!&s 'ds . &4'

We temporarily assume that the chain is inextensible, ex-pressed locally by the condition that % t!(s)%!1 everywhere.To get an energy functional generalizing Eq. &1', we note

that for a thin, homogeneous rod the elastic energy density isproportional to the square of the local curvature. Adding theexternal-force term from Eq. &1' yields

EWLC! t&s '$kBT

!&0

L totds' A2 (d t&s 'ds (2" f

kBTt&s '• z) . &5'

Equation &5' shows that parameter A is a measure of the bendstiffness of the chain. A is also the persistence length of thechain, the characteristic length scale associated with the de-cay of tangent-tangent correlations at zero stretching force:

( t&0 '• t&s ')WLC,e"%s%/A. &6'

The force-extension relation for the WLC was obtainednumerically in Ref. !6$; subsequently a high-precision inter-

FIG. 1. The freely jointed chain consists of identical segmentsof length b, joined together by free hinges. The configuration isfully described by the collection of orientation vectors " t i#. "- i#denotes the angle between t i and the fixed direction z of the appliedstretching force.

FIG. 2. A wormlike chain is a continuum elastic medium, whoseconfiguration is described in terms of the position vector r! as afunction of contour length s.

C. STORM AND P. C. NELSON PHYSICAL REVIEW E 67, 051906 &2003'

051906-2

FIG. 2: The freely-jointed chain model, consisting of N dis-crete sections of length b. From [2].

~t(s) =d~r(s)

ds, ~w(s) =

d~t(s)

ds. (3)

The inextensibility of the chain is enforced by the con-dition that |~t(s)| = 1 at all points along the chain [2].Noting that the energy of an elastic rod is proportionalto the square of the local curvature, and retaining theright-hand side of Eq. (1), the energy of the chain canbe expressed as [2]

EWLC [t(s)]

kBT=

∫ lc

0

(lp2

∣∣∣∣dt(s)ds

∣∣∣∣2 − f

kBTt(s) · z

)ds

(4)

where lc is the total “contour” length of the chain, and lpis the persistence length of the chain, which is the lengthscale at which directional correlations between chain seg-ments decay. Since this is not soluble exactly, interpo-lations of numerical approximations have yielded an ac-cepted functional form [3]

f =kBT

lp

[1

4

(1− z

lc

)−2

− 1

4+z

lc

]. (5)

This formula works well for f < 5 pN and lp � lc [3]. Atlow z, it reduces to an effective Hookean spring constantof

κWLC =3kBT

2lplc. (6)

2. EXPERIMENTAL SKETCH AND SALIENTDETAILS

The layout of the optical trap used in this experimentcan be seen in Fig. (4). The 975 nm near-infrared laseris collimated and directed through a 100x objective lens(OBJ), through the sample, and into a condenser lens

extension relation of the model. The fit value of b can thendepend both on the molecule under study and on its externalconditions such as salt concentration, as those conditions af-fect the intramolecular interactions.To formulate the FJC, we describe a molecular conforma-

tion by associating with each segment a unit orientation vec-tor t i , pointing in the direction of the ith segment, assketched in Fig. 1. In the presence of an external force f!along the z direction, we can define an energy functional forthe chain

E FJC!" t i#$

kBT!"%

i!1

N f bkBT

t i• z . &1'

In the absence of an external force, all configurations haveequal energy and &neglecting self-avoidance' the chain dis-plays the characteristics of a random walk. To pull the endsof such a chain away from each other a force has to beapplied, as extending the chain reduces its conformationalentropy. The resulting entropic elastic behavior can be sum-marized in the force-extension relation !3$

! zL tot" !coth# f b

kBT$"

kBTf b , &2'

the well-known Langevin function. In the limit of lowstretching force, all polymer models reduce to the Hooke-lawbehavior f!ksp(z); we define the effective spring constantby *!kspL tot , or

! zL tot"→

f*

#O& f 2'. &3'

Expanding Eq. &2' gives the effective spring constant for theFJC as *FJC!3kBT/b . The fact that the effective spring con-stant is proportional to the absolute temperature illustratesthat the elasticity in this model is purely entropic in nature.At high stretching force, Eq. &2' gives (z/L tot)→1; the

extension saturates when all the links of the chain are alignedby the external force. In reality, individual links are slightlyextensible; we will modify the model to introduce this effectin Sec. II C.

B. The wormlike chain

As mentioned above, double-stranded DNA &dsDNA' isfar from being a freely jointed chain. Thus it is not surprisingthat while the FJC model can reproduce the observed linearforce-extension relation of dsDNA at low stretching force,and the observed saturation at high force, still it fails at in-termediate values of f. Another indication that the model isphysically inappropriate is that the best-fit value of the Kuhnsegment length is b+100 nm, completely different from thephysical contour length per basepair of 0.34 nm.To improve upon the FJC, we must account for the fact

that the monomers do resist bending. In fact, the very greatstiffness of double-stranded DNA can be turned to our ad-vantage, as it implies that successive monomers are con-strained to point in nearly the same direction. Thus we cantreat the polymer as a continuum elastic body, its configura-tion described by the position r!(s) as a function of therelaxed-state contour length s &see Fig. 2'. Continuing totreat the chain as inextensible gives the wormlike chain !4,5$.The local tangent and curvature vectors ( t! and w! , respec-tively' are given by

t!&s '!dr!&s 'ds , w! &s '!

d t!&s 'ds . &4'

We temporarily assume that the chain is inextensible, ex-pressed locally by the condition that % t!(s)%!1 everywhere.To get an energy functional generalizing Eq. &1', we note

that for a thin, homogeneous rod the elastic energy density isproportional to the square of the local curvature. Adding theexternal-force term from Eq. &1' yields

EWLC! t&s '$kBT

!&0

L totds' A2 (d t&s 'ds (2" f

kBTt&s '• z) . &5'

Equation &5' shows that parameter A is a measure of the bendstiffness of the chain. A is also the persistence length of thechain, the characteristic length scale associated with the de-cay of tangent-tangent correlations at zero stretching force:

( t&0 '• t&s ')WLC,e"%s%/A. &6'

The force-extension relation for the WLC was obtainednumerically in Ref. !6$; subsequently a high-precision inter-

FIG. 1. The freely jointed chain consists of identical segmentsof length b, joined together by free hinges. The configuration isfully described by the collection of orientation vectors " t i#. "- i#denotes the angle between t i and the fixed direction z of the appliedstretching force.

FIG. 2. A wormlike chain is a continuum elastic medium, whoseconfiguration is described in terms of the position vector r! as afunction of contour length s.

C. STORM AND P. C. NELSON PHYSICAL REVIEW E 67, 051906 &2003'

051906-2

FIG. 3: The worm-like chain model, consisting of a continuousstrand with total length Ltot = lc. From [2].

(CD). The light is then directed into a quadrant pho-todetector (QPD). The QPD consists of four photodi-odes, and has two voltages as output: the sum of the twoupper voltages minus the sum of the two lower voltages,and the sum of the two left minus the two right. Thisprovides a measurement of the deflection of the light as itpasses through the sample, which is proportional to beaddisplacement within the trap.

The sample was visually imaged using a blue lamp.The light passes through the sample with orientation op-posite the laser, through a focusing lens, and then intoa CCD camera, which allows the sample to be visualizedon a computer screen.

The stage position was controlled in a rough mannerby hand-turned micrometers, and precisely by piezo ac-tuators. The piezo position and laser power were con-trolled within a MATLAB interface, which also handleddata collection. Calibrated strain gauges yielded a pre-cise measurement of the x-y stage position. The maxi-mum attainable laser power was 100 mW, however datawas acquired at 5 and 10 mW to avoid the application ofexcessive forces to the DNA strand.

2.1. Sample Preparation

DNA tethers were prepared through an involved seriesof stages (as in [3]) that attached 1.26 µm streptavidin-coated polystyrene microspheres to 3.5 kilobase-pair (kb)DNA, the other end of which was attached to a glasscoverslip. The DNA strand was prepared first by per-forming polymerase chain reaction (PCR) on a sequencewithin the M13mp18 plasmid. The process of PCR cansignificantly amplify a DNA sequence. First, the DNA isheated to denature the DNA into two strands; the tem-perature is then lowered so that primers may anneal tothe ends of the sequence by complimentary base-pairing.The temperature is then raised slightly to induce DNApolymerase to add complimentary nucleotides to extendthe sequence. This process was repeated 30 times us-ing an automated thermal cycler (a four-hour process),providing an amplification factor of approximately 230.The primers for one end of the DNA strand were func-tionalized with biotin, which has a high affinity for the

3

FIG. 4: Optical configuration of the trap, based on figurefrom [4].

FIG. 5: Flow channel between coverslip, slide, and double-stick tape. Based on figure from [4].

streptavidin molecules on the microspheres. The primersfor the opposite end were functionalized with digoxigenin,which binds to anti-digoxigenin antibodies later attachedto the coverslip. After the PCR reaction, agarose gelelectrophoresis confirmed that the product was 3.5kb inlength, and then the DNA was cleaned using a QiagenQiaQuick kit, which removed the DNA polymerase andunused nucleotides from the solution.

The glass coverslips were etched in a 1:1 solution ofpotassium hydroxide (KOH) and ethanol in order to

remove their wax-like coating. This allowed the anti-digoxigenin antibodies to bind to the surface. The glassslides were not etched, thus ensuring that only one sideof the sample would contain DNA tethers.

DNA-microsphere complexes were then prepared bymaking 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 stripsof double-stick tape side-by-side on a glass slide, whichwere then covered with an etched coverslip (Fig. 5).DNA tethers were then attached to the flow cells. First, a1:5 dilution of 20 mg/mL anti-digoxigenin in phosphate-buffered saline (PBS) solution was made. This was fur-ther diluted 1:10 in PBT, a solution containing PBS,bovine serum albumin, and Triton-X (a surfactant). A25 µL volume of the anti-digoxigenin solution was flowedinto the cell using a vacuum, and incubated for 40 min-utes at room temperature. Following this, 200 µL of a1 mg/mL casein (milk protein) in PBT solution, cleanedthrough a syringe filter, was flowed through the cell inorder to block the binding of other molecules to the cov-erslip. The sample was then incubated 20 minutes atroom temperature. Next, 25 µL of bead-DNA complexeswere flowed into the cell, and incubated for 20 minutesat room temperature. Finally, 800 µL of the casein so-lution was flowed through the cell to wash out unboundbeads. The sample was then sealed with vacuum greaseand immediately analyzed with the optical trap. Rapiddegradation of the samples necessitated the constructionof new ones for each day of data acquisition.

2.2. Trap Calibrations

The action of the optical trap on the 1.26 µmpolystyrene beads was characterized with automatedsoftware written by Dr. S. Wasserman. The Brown-ian motion of a bead in the trap was analyzed near thecoverslip because of the eventual measurement at thatlocation. A value of the trap stiffness α was found byanalyzing the power spectral distribution (PSD) of thefrequencies present in the Brownian oscillations.

Under the theory of Brownian motion, we expect thata particle in thermal equilibrium at temperature T willundergo random motion under what can be considered arandom force F (t). Since the force is random, we expectit to have a spectrum of white noise [1]. For a beadat low Reynolds number, as in this experiment, viscousdrag forces always dominate over the bead’s inertia; thismeans that we can express the force on the bead as afunction of time as

βx+ 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 themedium’s viscosity, and d is the diameter of the bead. A

4

PSD function can be defined using the Wiener-Khinchintheorem and the Fourier transform of the time-averagedautocorrelation function, following [1], leading to

Sxx(f) =

√kBT

π2β(f2 + f20 )(8)

in which f0 = α/2πβ. Thus, by examining the PSD ofa bead’s Brownian motion, the spring constant α of thetrap 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. Byusing the equipartition theorem, which holds that the en-ergy in a harmonic oscillator 1

2α〈x2〉 = 12kBT , the QPD

responsivity R (a conversion between QPD voltage andbead 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. Theconversion between stage position and strain gauge volt-age was provided by 20.309 staff as 2.22 V/µm.

2.3. DNA Tether Measurements

DNA tethers were identified by their highly-localizedBrownian motion. Once a bead was trapped, the stagewas oscillated at a low frequency (to minimize viscousdrag forces) and QPD displacement data was recorded.This provided a measurement of bead displacementwithin the trap (and thereby force) as a function of thelength to which the DNA was stretched. The measure-ment was optimized by adjusting the x-y stage positionto center the bead within the trap, performed by visu-ally observing the bead, as off-center beads oscillated in akinked path. The stage height was then adjusted in orderto position the bead as close to the coverslip as possibleby observing the QPD response: as the bead approachedthe coverslip, the maximum stretching distance increaseduntil the bead made contact, which then caused the datato lose its characteristic shape. The data was acquiredat a height just before this point.

3. DATA PRESENTATION AND ERRORANALYSIS

Five tethers were analyzed, at laser powers of 5 mWand 10 mW. The bead position was oscillated with fre-quency 0.55 Hz. QPD voltage as a function of straingauge voltage was recorded; an example plot of this datacan be seen in Fig. (6). The strain gauge voltage wasthen converted to stage position, and the QPD volt-age was converted to bead displacement within the trap.From these, the DNA extension z was calculated usingthe relation [3]

z =√h2 + (xstage − xbead)2 − r (9)

4 4.5 5 5.5 6 6.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

QPD Response vs Strain Gauge Stage Position

Strain Gauge Stage Position [V]

QP

D R

esponse V

oltage [V

]

FIG. 6: Raw data: QPD voltage (measure of bead deflection)versus strain gauge voltage (measure of stage position). Thecenter of the graph corresponds to zero extension. The curveshows the low-spring constant regime of the DNA (flatter cen-ter portion), the stretching as it approaches the strand length(steeper slope), and finally the bead being pulled out of thetrap (maximum/minimum).

0 100 200 300 400 500 600

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

DNA Restoring Force versus Tether Extension

Tether Extension [nm]

Resto

ring F

orc

e [pN

]

FIG. 7: DNA force-extension curve. Persistence Length:42.9 ± 0.7 nm, Contour Length: 812 ± 3 nm, χ2

r = 1.0.

which depends on the bead height h and the measuredquantities xstage and xbead (Fig. 8). The height h was

taken to be h = r since the beads were positioned directlyadjacent to the coverslip. The restoring force was calcu-lated by first converting QPD voltage to bead displace-ment with R, and then converting bead displacement totrapping force with α. A plot of restoring force versustether extension for one tether is visible in Fig. (7). Thedata has been binned into equally-spaced values of thetether extension; error bars result from the standard er-ror within each bin as well as from the calibrations.

5

FIG. 8: Diagram showing DNA extension z in relation tobead position. Here, xstage −xbead of Eq. (9) is representedby the quantity d.

Across the five tethers, the value of the DNA persis-tence length lp was measured as 53.0± 11.6 nm, and thecontour length lc was measured as 713 ± 36 nm. Withthe regime of small-displacement, the Hookean springconstant of DNA, following Eq. (6), is found to be0.162±0.035 pN/µm. The uncertainties reflect the stan-dard error of multiple measurements as well as the un-certainties from the fits.

4. DISCUSSION

Accepted values for lp are between 40-50 nm [3], andfor 3.5kb DNA, the accepted value of lc is approximately1180 nm [3]. Our agreement for the persistence lengthis quite satisfactory. The expected value for the small-displacement Hookean spring constant is 0.104 pN/µm;our result deviates by 1.6σ. Our measured value of thecontour length is low by 13σ, however, the order of mag-nitude is as expected. A possible explanation is the ex-treme sensitivity of the calibrations to the height of themicrosphere; however, without precise h measurementsit is difficult to quantify this. It is difficult to measure hprecisely because there is no piezoelectric control of the

stage height nor a way to accurately measure verticalbead deflection.

A further systematic uncertainty is the possible dam-age of the DNA strand due to multiple oscillations of thebead position. Care was taken to acquire data as soon aspossible once the stage was set to oscillate. This couldbe addressed in future iterations of this experiment bywriting a stage oscillation protocol that would find thecenter of DNA attachment and record QPD data withthe fewest possible number of oscillations. A possiblemethod for this is to sweep the bead in the y direction,identify the center, and then perform one extended sweepin the x direction.

Another cause of concern was the presence of smallforeign particles in the flow cells, which were occasionallydrawn into the trap during measurement. The data wasnot seen to fluctuate significantly when this occurred,however, in the future, more meticulous filtration of thesolutions may be desirable.

5. CONCLUSIONS

The primarily entropic spring-like properties of DNAwere observed by manipulation of a single molecule in anoptical trap. With a fit to a worm-like chain model, thepersistence and contour lengths of a 3.5kb DNA strandwere measured to be 53.0±11.6 nm and 713±36 nm, re-spectively. The former is in agreement with the acceptedrange of 40-50 nm, while the latter, despite being correctin order of magnitude, deviates by 13σ. The Hookeanspring constant of DNA, at small extension distances,was found to be 0.162 ± 0.035 pN/µm, a 1.6σ deviationfrom the expected value. Future extensions of the exper-iment could test different stretching regimes of DNA orother force-extension models, such as those that includecorrections for aqueous ion interactions or enthalpic prop-erties [5].

[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 Jour-

nal 72, 1335 (1997).

Acknowledgments

The author gratefully acknowledges Ariana Mann,Gustaf Downs, and Devin Cela for their equal part in per-

forming this experiment, which included many full daysof preparation. The author would also like very much tothank Dr. Steve Wasserman and Dr. Steven Nagle whowere invaluable at every stage of this experiment.