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Pro gradu tutkielma Fysiikan suuntautumisvaihtoehto STRETCHING SINGLE DNA-MOLECULES WITH TEMPERATURE-STABILIZED OPTICAL TWEEZERS Antti Rahikkala 29.12.2009 Ohjaajat: FM Anders Wallin, Ph.D. Gabija Ziedaite Tarkastajat: Prof. Edward Hæggström Ph.D. Imad Abbadi HELSINGIN YLIOPISTO FYSIIKAN LAITOS PL 64 (Gustaf Hällströmin katu 2) 00014 Helsingin yliopisto

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Pro gradu –tutkielma

Fysiikan suuntautumisvaihtoehto

STRETCHING SINGLE DNA-MOLECULES WITH TEMPERATURE-STABILIZED OPTICAL TWEEZERS

Antti Rahikkala

29.12.2009

Ohjaajat: FM Anders Wallin, Ph.D. Gabija Ziedaite

Tarkastajat: Prof. Edward Hæggström

Ph.D. Imad Abbadi

HELSINGIN YLIOPISTO

FYSIIKAN LAITOS

PL 64 (Gustaf Hällströmin katu 2) 00014 Helsingin yliopisto

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HELSINGIN YLIOPISTO HELSINGFORS UNIVERSITETTiedekunta/Osasto Fakultet/Sektion Faculty/SectionMatemaattis-luonnontieteellinen tiedekunta

Laitos Institution DepartmentFysikaalisten tieteiden laitos

Tekijä Författare AuthorRahikkala, Antti Tuomas Antero

Työn nimi Arbetets title TitleStretching Single DNA Molecules with Temperature Stabilized Optical Tweezers

Oppiaine Läroämne SubjectFysiikka

Työn laji Arbetets art LevelPro gradu -tutkielma

Aika Datum Month and yearjoulukuu 2009

Sivumäärä Sidoantal Number of pages37+8

Tiivistelmä – Referat AbstractTämä tutkielma koostuu kahdesta osasta; ensimmäisessä osassa tutkimme optinen pinsetti-laitteistomme kalibraatiota ja soveltuvuutta yksittäismolekyylikokeisiin 10kb pituisien λ-faagista peräisin olevien DNA-molekyylien voima-venytyskokeilla. Mittauksiin sovitettiin ”worm-like chain”-interpolaatio malli, joka osoitti, että ~71%:lla löydetyistä DNA ketjuista oli pituus 15% sisällä odotetusta pituudesta (3.38 µm). Vain 25%:lla DNA:sta oli sitkeyspituus 30-60 nm odotetun arvon ollessa 40-60 nm. Tutkielman toisessa osassa rakensimme lämpötilakontrollerin, jonka tarkoituksena oli poistaa lämpötilavaihteluiden aiheuttama ”ajelehtiminen” optisilla pinseteillä kiinnipidetyn mikroskooppisen pallonpaikassa. Kontrolleri käyttää ”feedforward-” ja takaisinkytkentäsilmukoita saavuttaakseen

1.58 mK sisäisen tarkkuuden ja 0.3 K ulkoisen tarkkuuden. Viiden minuutin kokeen aikana pallo ajelehti 1.4 nm/min avoimella silmukalla ja 0.6 nm/min suljetulla silmukalla.

Avainsanat – Nyckelord Keywordsoptiset pinsetit, lämpötila kontrolleri, DNA venytys, yksittäismolekyylibiologia

Säilytyspaikka – Förvaringställe Where depositedKumpulan tiedekirjasto

Muita tietoja Övriga uppgifter Additional information

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HELSINGIN YLIOPISTO HELSINGFORS UNIVERSITETTiedekunta/Osasto Fakultet/Sektion Faculty/SectionFaculty of Science

Laitos Institution DepartmentDepartment of Physical Sciences

Tekijä Författare AuthorRahikkala, Antti Tuomas Antero

Työn nimi Arbetets title TitleStretching Single DNA Molecules with Temperature Stabilized Optical Tweezers

Oppiaine Läroämne SubjectPhysics

Työn laji Arbetets art LevelMaster‟s Thesis

Aika Datum Month and yearDecember 2009

Sivumäärä Sidoantal Number of pages37+8

Tiivistelmä – Referat AbstractThis thesis consists of two parts; in the first part we performed a single-molecule force extension measurement with 10kb long DNA-molecules from phage-λ to validate the calibration and single-molecule capability of our optical tweezers instrument. Fitting the worm-like chain interpolation formula to the data revealed that ~71% of the DNA tethers featured a contour length within ±15% of the expected value (3.38 µm). Only 25% of the found DNA had a persistence length between 30 and 60 nm. The correct value should be within 40 to 60 nm. In the second part we designed and built a precise temperature controller to remove thermal fluctuations that cause drifting of the optical trap. The controller uses feed-forward and PID (proportional-integral-derivative) feedback to achieve 1.58 mK precision and 0.3 K absolute accuracy. During a 5 min test run it reduced drifting of the trap from 1.4 nm/min in open-loop to 0.6 nm/min in closed-loop.

Avainsanat – Nyckelord Keywordsoptical tweezers, temperature control, DNA stretching, single molecule biology

Säilytyspaikka – Förvaringställe Where depositedKumpula Science Library

Muita tietoja Övriga uppgifter Additional information

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Table of contents

1. Introduction .................................................................................................................. 5 2. Theory ........................................................................................................................... 7

2.1 Optical tweezers ................................................................................................... 7 2.2 The Freely-jointed chain and the worm-like chain models .................................... 8

2.2.1 Freely-jointed chain .............................................................................................. 8 2.2.2 Worm-like chain ................................................................................................. 10

2.3 Temperature controller for the objective of the optical tweezers ....................... 14 2.3.1 Temperature measurement circuit ...................................................................... 14 2.3.2 Temperature control circuit ................................................................................ 16 2.3.3 Feed-forward- and feedback controller ............................................................... 17

3. Materials and methods .............................................................................................. 19

3.1 Preparation of biotin-digoxigenin labelled DNA by PCR .................................... 19 3.2 Preparation of dumbbell assays .............................................................................. 20

3.2.1 Streptavidin beads ............................................................................................... 20 3.2.2 Protein G beads ................................................................................................... 21 3.2.3 The sample mix ................................................................................................... 21

3.3 Laminar flow chambers .......................................................................................... 22

3.4 Calibration of OT ..................................................................................................... 22 3.4.1 Calibration of position sensitive detectors .......................................................... 22 3.4.2 Calibration of trap stiffness ................................................................................. 23

3.5 Stretching DNA ........................................................................................................ 25 3.6 Building the temperature controller ...................................................................... 25

3.6.1 Attaching the Pt100 transducer and the heating-wire to the objective ............... 25 3.6.2 Manufacturing of temperature-measurement and heater circuits ....................... 27

3.6.3 Tuning the feed-forward feedback loop .............................................................. 27 3.6.4 LabVIEW 8.6 program and DAQ-card with temperature control ...................... 28

4. Results ......................................................................................................................... 29 4.1 Contour- and persistence lengths of 10kb DNA .................................................... 29 4.2 Testing the temperature controller ........................................................................ 31

4.2.1 The time constants of the Pt100 .......................................................................... 31 4.2.2 The cutoff frequency of the 2

nd order Sallen-Key low-pass filter ...................... 31

4.2.3 Step tests on the temperature controller .............................................................. 33 4.2.4 Long-term closed-loop stability .......................................................................... 34

4.2.5 Long-term open-loop stability ............................................................................ 35 4.2.6 Power spectral density ........................................................................................ 35 4.2.7 Optical tweezers trap stability ............................................................................. 36

5. Discussion ................................................................................................................... 37

6. Conclusions ................................................................................................................. 37 References ........................................................................................................................... 38 Appendix ............................................................................................................................. 41

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1. Introduction

Deoxy-ribonucleic acid, or DNA, the code of life, is a molecule that is made of two long

polymers. The polymers themselves are based on nucleotides, simple molecules, which

consist of a nucleobase (referred to as a base), a five-carbon sugar (2‟-deoxyribose), and

one to three phosphate groups. DNA contains all the genetic information required to

construct cells and proteins for each living creature. DNA was first discovered by a Swiss

physician Friedrich Miescher in 1869 [1], however he could not decipher its function.

Although earlier suggested [2], finally in 1952 the hereditary function of DNA was proven

by Alfred Hershey and Martha Chase as they found that DNA is the genetic material of the

phage T2 [3]. A year later, James D. Watson and Francis Crick presented the correct

double-helix model of DNA structure [4]. Double-stranded DNA is ~2 nm in diameter and

the haploid human genome is ~1 m in length [5]. The length of a DNA molecule can also

be reported as the number of base pairs it contains. For example, the human haploid

genome, stored on 23 chromosomes, contains ~3 billion base pairs [6]. DNA is a

polymorphic molecule. There are three biologically active forms of DNA: A-, B-, and Z-

DNA [7]. Geometrically, A- and B-DNAs are right-handed. The difference is that the A-

DNA has a more compactly packed helical structure. The Z-DNA form is left-handed.

However, the only DNA occurring naturally is B-DNA, the other forms (A and Z) being

results of artificial changes in the cell environment [7].

Chromosome segregation in dividing cells and packaging of nucleic acids (DNA/RNA)

into viral capsids involves individual molecular motors generating forces of ~50 pN [8].

These molecular motors are proteins that are the smallest machines known. They provide

forces involved in biological motion. For example, muscles can generate forces of ~50 N

due to collective action of myosin motors (<50 nm in size). However, a single myosin

motor is capable of only ~5 pN force [9]. DNA/RNA-binding enzymes and molecular

motors are present everywhere in the biosphere. Many diseases may be traced back to

dysfunction of these enzymes. Furthermore, this knowledge of their function helps to

understand and treat diseases such as genetic disorders and cancer [10]. Knowledge of how

chemical reactions are transformed into mechanical force by molecular motors may allow

human-made nano-scale machine fabrication.

Previously both DNA and molecular motors were studied in bulk by biologists and

biochemists [11]. However, recent development in instrumentation has allowed

experimenting on single molecules. Capability to observe e.g. single proteins in action

gives more information than bulk experiments. For example, many proteins exist in

different states e.g. folded, unfolded, or an intermediate transition state. In bulk

experiments the latter is hardly observable; however in single-molecule (SM) experiments

one can accurately characterize such a transition [11]. Furthermore, a fluorophore molecule

can be attached to a protein and using equipment capable of detecting fluorescence one can

observe a single protein on its way along a strand of DNA [11]. Ability to obtain

characteristics of single molecules allows analyzing the mechanisms of their action and

testing the models statistically.

Suitable instruments for SM experiments must be capable of manipulating single micron-

sized objects that are processed such that molecules of interest can attach to them, measure

forces of picoNewton scale, and have nanometer scale spatial resolution. These

instruments include atomic force microscopes (AFM) [11], magnetic tweezers (MT) [11],

optical tweezers (OT), and the bio-membrane force probe (BMFP) [11]. The AFM has a

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dual role in SM experiments. Firstly, it is capable of imaging the molecules of interest and

secondly, by chemically coating the tip single molecules can be attached between it and the

surface of the measurement chamber. AFM can measure forces in the range of 20 pN-10

nN. The MT uses two magnets to generate a magnetic field that can be used to manipulate

micron-sized beads of magnetic material. The studied molecule can be attached between

the bead and the chamber wall or a micropipette. The useful force region of MT is 10-2

-10

pN. Worth noting is that MT is capable of twisting the bead allowing experiments of

torsional properties of molecules [11, 12]. The OT uses the optical gradient force of a

focused beam of light to generate forces in the range of 0.1-100 pN onto micron-sized

dielectric beads. The bead‟s index of refraction has to be higher than that of the

surrounding medium. A molecule can be attached between the bead and the chamber wall

or a micropipette, or between two beads if the instrument has two trap beams. The OT can

also be used in torque measurements [13]. The BMFP is a method where a cell (e.g. red

blood cell) is attached between a micropipette and a bead held tightly with another

micropipette. This can be used to measure membrane tensions [11].

In our laboratory the instrument of choice to study single biological molecules is the

optical tweezers [14]. First introduced by Ashkin [15, 16], the optical tweezers offer easily

adjustable trap stiffness (laser power and a choice of bead diameter), no micropipettes are

needed, and the force range is suitable for experiments with DNA, RNA, and molecular

motors, which are of interest in our laboratory. The drawback of OT is its susceptibility to

trap drift due to mechanical and acoustic vibrations, air currents where the laser propagates

(on the optical table), and thermal fluctuations in components. The vibrations can be

countered by isolating the system form the environment, the air currents by replacing the

air atmosphere of the sealed optical table with helium [17], and controlling temperature in

components that absorb power from the beam of light [18]. The beads can be manipulated

inside microfluidic channels [19] capable of micro-liter sample handling.

Before one can study more complex molecules, the required skills must be mastered with

simpler molecules and the system must be validated with a „benchmark‟ molecule. One

such commonly [20-22] used molecule is phage-λ DNA. This thesis consists of two parts:

in the first part force-extension measurements on a 10kb λ-phage biotin-digoxigenin DNA

construct were performed in order to validate the calibration and SM suitability of our

instrument. Results are fit to the extensible worm-like chain model and the contour- and

persistence length of the construct are extracted. In the second part a temperature controller

for the trapping objective was designed, built, and integrated into our instrument to reduce

trap drifting due to thermal fluctuations. The results show that the drift was reduced by

~50%, from 1.4 nm/min to 0.6 nm/min during a 5 minute measurement. This will directly

benefit e.g. force-clamp experiments that require holding the beads steady for long times

[23, 24].

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2. Theory

2.1 Optical tweezers

The optical tweezers instrument in our laboratory [25] uses a 1064 nm laser to trap and

control dielectric beads of micron scale diameter. The laser is divided to produce one static

and one steerable trap. This method is cheaper and simpler than a system consisting of dual

steerable traps and both are equally qualified for force-extension measurements. Two

detection lasers (785 and 830 nm, HL7851G, Hitachi, Tokyo, Japan, and DL5032-001,

Thorlabs, Newton, NJ) measure the position of the trapped beads. The manipulation of the

beads takes place inside laminar flow chambers (see sec. 3.3), into which the lasers are

focused using a high-NA 100x microscope objective (TIRF 100x, 1.49 N.A., Nikon,

Tokyo, Japan). Near the focus, a bead feels an attractive force towards the focus due to the

intensity gradient. The beam acts as a spring, exerting a Hookean force on the trapped

bead. The schematic of the OT is presented in Fig. 1.

Figure 1. The schematic of the optical tweezers. The red dashed lines indicate the Back-Focal-Plane (BFP) and its

conjugate planes.

Lenses L1 and L2 collimate the trap laser (Compass-4000, 1064 nm, Coherent, Santa

Clara, CA) before it goes through a Faraday isolator (FI, FI-1060-5SI, Linos, Goettingen,

Germany) that prevents beam reflections from re-entering the laser. The shutter (SH1,

SH05, Thorlabs) can be used to disable or enable the laser. A half-wave plate (HWP1)

rotates the polarization of the beam prior to splitting by a polarizing beam splitter (PBS1,

05BC16PC.9, Newport, Irvine, CA). The s-polarized part, reflected 90-degrees in PBS1,

forms the static trap, and the p-polarized part, transmitted through PBS1, forms the

steerable trap. Two acousto-optic deflectors (AOD, 45035-3-6.5deg-1.06-xy, NEOS-

Technologies, Melbourne, FL) control the position of the steerable trap. Both traps can be

enabled and disabled via software, the static trap by a shutter (SH2, same model as SH1)

and the steerable trap by switching on/off the power to the AOD. The polarizing beam

splitter (PBS2, same model as PBS1) combines the trap lasers after which lenses L3 and

L4 expand them three-fold to prepare them to be merged with the detector beams. The

merging is done with a dichroic mirror D1 (SWP-45-RU1064-TU850-PW-2025-C, CVI

Laser LLC, Albuquerque, NM) that reflects wavelengths of ~1064 nm and transmits other

wavelengths.

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The detection lasers (830 & 785) pass through half-wave plates (HWP 2&4) that rotate the

polarization so that the laser intensity can be adjusted as they enter Faraday isolators (FI,

DLI-1, Linos) that prevent optical feedback. The HWPs 3&5 rotate the polarization so that

the beam enters the single-mode fiber (SMF) with a polarization direction that matches the

polarization maintaining axis [26] in the SMF. The beam that exits the fiber features the

TM00 mode [27] and is collimated with an optical coupler (OC). A polarizing beam splitter

PBS3 (PBSH-450-1300-050, CVI Melles Griot, Albuquerque, NM) then combines the

beams from both the detector lasers, after which they enter the microscope. Lenses L7 and

L8 expand the detection beams and image the BFP such that adjusting the mirrors in front

of PBS3 allows steering the beams in the sample-plane.

Both the trapping and detection beams are reflected by D2 (780dcspxr, Chroma,

Brattleboro, VT, USA). Lenses L5 and L6 form a telescope that relays the BFP-image to

allow trap-steering and also allow focus adjustment of the trap. The lasers enter the back

focal plane (BFP) of the objective and focus into the sample before the condenser (COND,

T-C High NA (oil), Nikon) collimates them. The dichroic mirror D3 (TLM2-800-45-UNP-

2037, CVI Laser LLC) works similarly as D2, reflecting the detector beams towards the

detectors (PD 1&2). The filter (F1, FM203, Thorlabs) prevents visible light from reaching

the detectors. PBS4 (10FC16PB.5, Newport) separates the detector lasers into s- and p-

polarized beams. The former is reflected and propagates through F2 (FL830-10, Thorlabs),

which passes 830 nm, whereas the latter goes through F3 (LL01-785, Semrock, Rochester,

NY, USA), which passes 785 nm. The position detector PD1 (S2-0171, Sitek, Partille,

Sweden) measures the center of intensity of the 830 nm beam and PD2 (same as PD1) does

the same for the 785 nm beam. A green LED (LED) illuminates the sample plane. The

beam from the LED passes through the objective and a pair of filters (F4&5, KG1 and

KG3, Schott, Mainz, Germany), which remove the remaining intensity of the lasers that

might get through D2. Thus, the LED provides the bright-field picture we see on a monitor

via the CCD camera.

2.2 The Freely-jointed chain and the worm-like chain models

Two theories of force extension behaviour of long polymers have been developed: the

freely jointed chain (FJC) [28] and the worm-like chain (WLC) [29] models. Before

moving to more elaborate single-molecule experiments with DNA and proteins, a simple

DNA stretching experiment is widely used as a proof-of-principle measurement. To verify

the results from a stretching experiment, a model of DNA stretching behaviour must be

fitted into the data. Here we will derive both the models.

2.2.1 Freely-jointed chain

The FJC model treats the polymer as if it was divided into N segments that are freely

jointed to each other (Fig. 2). These segments are also known as Kuhn segments [5]. As a

result, in equilibrium the DNA can be modelled as a random walk of these segments [5],

i.e. the orientation of a segment is independent of the orientation of its adjacent segments.

When a force is exerted on the DNA, the segments tend to align in the direction of the

force. This stretching is opposed by the tendency of the polymer to maximize its entropy.

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Figure 2. A 3D image of a Kuhn segment b. The molecule is stretched at direction of the force. The projection of the segment at the force direction is given by bf. Note direction of the Cartesian and spherical coordinate systems used in the text.

In thermal equilibrium the expected length of the projection of the segment in the force direction is given by

cos( )22

0 0cos( )2

2

0 0

cos( ) sin( )

sin( )

B

B

Fbk T

F Fbk T

b b e d db

b e d d

cos( )

cos( )

cos( ) sin( ) k Tb b e d dcos( ) sin( )b b e d dcos( ) sin( ) Bb b e d dBk Tb b e d dk TBk TBb b e d dBk TB

k Tb e d dBb e d dBk Tb e d dk TBk TBb e d dBk TB

0 02

2

cos( ) sin( )b b e d dcos( ) sin( )b b e d dcos( ) sin( )2cos( ) sin( )b b e d d2b b e d d2cos( ) sin( )b b e d dcos( ) sin( )2cos( ) sin( )2b b e d d2cos( ) sin( )2

22 sin( )b e d d2b e d d2 sin( )b e d dsin( )sin( )b e d dsin( )b e d dsin( )

,

(1)

where cos( )bcos( ) is the projection in the direction of the force, 2 sin( )b sin( ) comes from integration in spherical coordinates. The exponential term is a Boltzmann factor that gives, in a system of many possible states, the relative probability of the state with energy

cos( )Fbcos( )cos( )Fbcos( ) in thermal equilibrium, where the term cos( )Fbcos( )cos( )Fbcos( ) is the potential energy of the segment that is aligned with the external force F. The radius (Kuhn segment b) in this spherical integration is constant, so integration over it is not performed. Since there is no dependency on φ in Eq. (1), integration over it yields 2π. This gives

cos( )2

0cos( )

2

0

cos 2 sin

2 sin

B

B

Fbk T

F Fbk T

b b e db

b e d

cos( )

cos( )

k Tb b e dBb b e dBk Tb b e dk TBk TBb b e dBk TB

k Tb e dBb e dBk Tb e dk TBk TBb e dBk TB

2cos 2 sinb b e d2b b e d2cos 2 sinb b e dcos 2 sin2cos 2 sin2b b e d2cos 2 sin2

0

cos 2 sinb b e dcos 2 sinb b e dcos 2 sincos 2 sinb b e dcos 2 sinb b e dcos 2 sincos 2 sinb b e dcos 2 sinb b e dcos 2 sin

2 sin22 sinb e d2b e d22 sinb e d2 sin22 sin2b e d22 sin2

,

(2)

and integration over θ yields

coth BF

B B

k TFb Fbb b bk T Fb k T

k TFb Fbk TFb Fbk Tb b bFb Fbk TFb Fbk Tcothb b bcothb b bcothb b bcothb b bcoth Fb Fbk TFb Fbk Tb b bb b bb b bBb b bBb b bBb b bBFb Fbk TFb Fbk TBk TBFb FbBk TBb b bBb b bBFb Fbb b bFb FbBFb FbBb b bBFb FbBk TFb Fbk Tb b bk TFb Fbk TBk TBFb FbBk TBb b bBk TBFb FbBk TB

k T Fb k TB Bk T Fb k TB Bk T Fb k TB Bk T Fb k Tk T Fb k Tk T Fb k T,

(3)

where is the Langevin function 1( ) cothx x x1( ) cothx x( ) cothx x( ) coth x [30]. Since a polymer is made up

of N segments, its average end-to-end distance R(F) is

( )B B

Fb FbR F Nb Lk T k TFb FbR F Nb LFb FbR F Nb LR F Nb LFb FbR F Nb LFb FbR F Nb LFb Fb

B Bk T k TB Bk T k TB Bk T k Tk T k T,

(4)

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where L=Nb is the contour length of the DNA, i.e. the length of the molecule when it is fully stretched [5]. The Laurent series [31] of coth a is

22 1 32

1

21 1 1 1coth ...,(2 )! 3 45

nnn

n

Ba a a aa n a

221 1 1 121 1 1 1221 1 1 12 n1 1 1 1n1 1 1 11 1 1 1B1 1 1 11 1 1 11 (2 )! 3 45a n a1a n a1 (2 )! 3 45a n a(2 )! 3 45

2 1 31 1 1 12 1 31 1 1 12 1 3coth ...,2 1 3coth ...,2 1 32 1 31 1 1 12 1 3coth ...,2 1 31 1 1 12 1 3

(2 )! 3 45coth ...,

(2 )! 3 45coth ...,coth ...,a a a acoth ...,coth ...,a a a acoth ...,coth ...,a a a acoth ...,coth ...,a a a acoth ...,1 1 1 12 1 31 1 1 12 1 3coth ...,2 1 3coth ...,2 1 32 1 31 1 1 12 1 3coth ...,2 1 31 1 1 12 1 3coth ...,a a a acoth ...,1 1 1 1coth ...,1 1 1 1coth ...,1 1 1 1coth ...,coth ...,a a a acoth ...,coth ...,a a a acoth ...,1 1 1 11 1 1 1B1 1 1 11 1 1 121 1 1 1221 1 1 121 1 1 1n1 1 1 11 1 1 1B1 1 1 11 1 1 1

(2 )! 3 45a n a(2 )! 3 45a n a(2 )! 3 45a n a(2 )! 3 45a n a(2 )! 3 451 1 1 12 1 31 1 1 12 1 321 1 1 121 1 1 121 1 1 12coth ...,2 1 3coth ...,2 1 32coth ...,21 1 1 1coth ...,1 1 1 12 1 31 1 1 12 1 3coth ...,2 1 31 1 1 12 1 321 1 1 12coth ...,21 1 1 1221 1 1 12coth ...,21 1 1 12 2 1 31 1 1 12 1 3n2 1 31 1 1 12 1 32 1 3coth ...,2 1 3n2 1 3coth ...,2 1 32 1 31 1 1 12 1 3coth ...,2 1 31 1 1 12 1 3n2 1 31 1 1 12 1 3coth ...,2 1 31 1 1 12 1 3coth ...,ncoth ...,1 1 1 1coth ...,1 1 1 1n1 1 1 1coth ...,1 1 1 1coth ...,

(2 )! 3 45coth ...,

(2 )! 3 45coth ...,1 1 1 1B1 1 1 121 1 1 12B21 1 1 121 1 1 1coth ...,1 1 1 1B1 1 1 1coth ...,1 1 1 121 1 1 12coth ...,21 1 1 12B21 1 1 12coth ...,21 1 1 12coth ...,a a a acoth ...,coth ...,a a a acoth ...,2coth ...,2a a a a2coth ...,2coth ...,ncoth ...,a a a acoth ...,ncoth ...,1 1 1 121 1 1 121 1 1 1B1 1 1 11 1 1 12 1 31 1 1 12 1 3coth ...,2 1 3coth ...,2 1 32 1 31 1 1 12 1 3coth ...,2 1 31 1 1 12 1 3coth ...,a a a acoth ...,

(5)

where Bn is a Bernoulli number [31]. At low forces, Fb<<kBT, we can omit higher powers of Fb/kBT to get

coth ,3 3

B B B

B B B

k T k T k Tz Fb Fb FbL k T Fb Fb k T Fb k T

k T k T k Tz Fb Fb Fbk T k T k Tz Fb Fb Fbk T k T k Tcoth ,coth ,coth ,coth ,coth ,coth ,B B Bcoth ,coth ,z Fb Fb Fbcoth ,z Fb Fb Fbcoth ,coth ,B B Bcoth ,z Fb Fb Fbcoth ,B B Bcoth ,k T k T k Tz Fb Fb Fbk T k T k Tcoth ,k T k T k Tcoth ,z Fb Fb Fbcoth ,k T k T k Tcoth ,B B Bk T k T k TB B Bz Fb Fb FbB B Bk T k T k TB B Bcoth ,B B Bcoth ,k T k T k Tcoth ,B B Bcoth ,z Fb Fb Fbcoth ,B B Bcoth ,k T k T k Tcoth ,B B Bcoth ,z Fb Fb Fbcoth ,z Fb Fb Fbcoth ,L k T Fb Fb k T Fb k T

coth ,L k T Fb Fb k T Fb k T

coth ,3 3L k T Fb Fb k T Fb k T3 3

coth ,3 3

coth ,L k T Fb Fb k T Fb k T

coth ,3 3

coth ,L k T Fb Fb k T Fb k T

coth ,L k T Fb Fb k T Fb k T

coth ,z Fb Fb Fbk T k T k Tz Fb Fb Fbk T k T k Tcoth ,coth ,coth ,coth ,coth ,B B Bcoth ,coth ,coth ,B B Bcoth ,z Fb Fb Fbcoth ,z Fb Fb Fbcoth ,coth ,B B Bcoth ,z Fb Fb Fbcoth ,B B Bcoth ,k T k T k Tz Fb Fb Fbk T k T k Tcoth ,k T k T k Tcoth ,z Fb Fb Fbcoth ,k T k T k Tcoth ,B B Bk T k T k TB B Bz Fb Fb FbB B Bk T k T k TB B Bcoth ,B B Bcoth ,k T k T k Tcoth ,B B Bcoth ,z Fb Fb Fbcoth ,B B Bcoth ,k T k T k Tcoth ,B B Bcoth ,B B BL k T Fb Fb k T Fb k TB B BL k T Fb Fb k T Fb k TB B BL k T Fb Fb k T Fb k T

coth ,L k T Fb Fb k T Fb k T

coth ,L k T Fb Fb k T Fb k T

coth ,(6)

where zL

is the extension in relation to the contour length of the molecule in the direction

of the force such that 0< zL

≤1. This yields the low force approximation

3 .Bk T zFb L

z3 Bk TBk TB

b Lzzz

b L.

b Lb L(7)

Thus, at low forces the freely-jointed chain may be treated as a Hookean spring with a spring constant / 3 / 3 / 2B BF z k T b k T P/ 3 / 3 / 2B B/ 3 / 3 / 2B B/ 3 / 3 / 2F z k T b k T P/ 3 / 3 / 2F z k T b k T P/ 3 / 3 / 2/ 3 / 3 / 2B B/ 3 / 3 / 2F z k T b k T P/ 3 / 3 / 2B B/ 3 / 3 / 2/ 3 / 3 / 2F z k T b k T P/ 3 / 3 / 2F z k T b k T P/ 3 / 3 / 2 , where P is the persistence length (half the Kuhn length, i.e. 2P=b) that describes the bending stiffness of the molecule, i.e. thelength at which the other end of a strand is not affected if the other end is bent [5]. Thus the FJC provides two approximations: Eq. (7) can be used at low forces (Fb<<kBT), while the exact expression, Eq. (4), performs well at low and high forces (z/L>0.9).

2.2.2 Worm-like chain

The WLC model considers DNA to be rod-shaped, inextensible, and made of an isotropic elastic material characterized by its Young‟s modulus and the rod‟s second moment of inertia (the rod‟s resistance to bending and deflection) [5]. This model better describes the DNA molecule‟s entropic elasticity, as it includes small thermal fluctuations along the

molecular axis. P, the persistence length, is dependent on the salt concentration: at 150 mMNaCl the persistence length for DNA is ~50 nm [32]. To align and straighten the elastic units, forces of the order of /Bk T P are needed. For larger forces, /BF k T P/BF k T P/F k T P/BF k T PB , the effective energy Eelastic of a stretched WLC molecule can be used to approximate the force-extension behaviour [33]

22

0

1 ( )ˆ

2

Lelastic

sB

E r sEI ds Fzk T s

21 ( )

221 ( )1 ( )21 ( )21 ( )k T s2k T s2

EI ds FzEI ds FzEI ds FzEI ds FzEI ds FzEI ds Fzk T sk T s

1 ( )1 ( )L1 ( )k T s0k T s0

1 ( )1 ( )21 ( )21 ( )r s1 ( )1 ( )EI ds Fz1 ( )EI ds Fz1 ( )k T s

1 ( )1 ( )1 ( )r s1 ( )EI ds FzEI ds Fz1 ( )EI ds Fz1 ( )1 ( )1 ( )EI ds Fz1 ( )EI ds Fz1 ( )EI ds Fz1 ( )1 ( )1 ( )r s1 ( )EI ds Fz1 ( )EI ds Fz1 ( )1 ( )r s1 ( )EI ds Fz1 ( )r s1 ( )EI ds Fzk T sk T sk T sk T ssk T ssk T s

.(8)

Here ( )r s is the position vector along the chain, z the unit vector in the direction of the force (see Fig. 3) F, E is the Young‟s modulus, and I is the second moment of inertia [34] around an axis of the cross-section.

Page 11: Stretching Dna

11

Figure 3. The vectors used with WLC model.

The unit tangent vector along the chain, ˆ( )t s , can be defined as ( )ˆ( ) r st s

s( )r s( )r s( )( )r s( )r s( )s

and we get

2

2

ˆ( )ˆ( )z

r s tk t z ts s s

2 ˆ( )r s t( )r s t( ) ( )k t z tk t z tk t z tk t z t( )k t z t( )( )( )k t z t( )ˆ( )r s t( )r s t( )k t z t( )k t z t( ) ( )k t z t( )ˆ( )ˆk t z tˆ( )ˆr s tk t z tr s t( )r s t( )k t z t( )r s t( )

2 zs s s2 ( )z( )z( )k t z t2k t z t2 ( )k t z t( )( )z( )k t z t( )z( )( )( )k t z t( ) ,(9)

where [ , ]x yt t tx yt t t[ , ]x y[ , ]x y[ , ]t t t[ , ]t t t[ , ][ , ]x y[ , ]t t t[ , ]x y[ , ][ , ]x y[ , ]x y[ , ]t t t[ , ]t t t[ , ][ , ]x y[ , ]t t t[ , ]x y[ , ] . Using the Taylor expansion the tangent vector in z-direction can be expressed as

22 41 1 ( ) ...

2ztt t t2

2 41 1 ( ) ...2 41 1 ( ) ...2 41 1 ( ) ...t2 4t2 42 41 1 ( ) ...2 4t2 41 1 ( ) ...2 4t t tt t tt t tt t t1 1 ( ) ...t t t1 1 ( ) ...2 41 1 ( ) ...2 4t t t2 41 1 ( ) ...2 42 42 41 1 ( ) ...2 41 1 ( ) ...2 41 1 ( ) ...t t t1 1 ( ) ...2 41 1 ( ) ...2 4t t t2 41 1 ( ) ...2 42 42 41 1 ( ) ...2 4t2 4t2 42 41 1 ( ) ...2 4t2 41 1 ( ) ...2 41 1 ( ) ...t t t1 1 ( ) ...2 41 1 ( ) ...2 4t t t2 41 1 ( ) ...2 42 42 41 1 ( ) ...2 41 1 ( ) ...t t t1 1 ( ) ...2 41 1 ( ) ...2 4t t t2 41 1 ( ) ...2 41 1 ( ) ...2

1 1 ( ) ...2

1 1 ( ) ...1 1 ( ) ...t t t1 1 ( ) ...1 1 ( ) ...t t t1 1 ( ) ...1 1 ( ) ...1 1 ( ) ...t t t1 1 ( ) ...1 1 ( ) ...t t t1 1 ( ) ...1 1 ( ) ...t t t1 1 ( ) ...1 1 ( ) ...t t t1 1 ( ) ...(10)

Now, dropping terms of order 4( )t 4( )4( )4( )t( )( )( )t( ) and higher, i.e. assuming that the chain is extended ( zt tt tt t ), the equation for k becomes

2

ˆ12

tk z ts

22tk z tk z tk z tttk z tk z tk z tttk z tk z t1k z t1k z tk z tk z tk z tk z tˆk z tˆk z tˆk z tˆk z tˆk z t1k z t1k z tk z t1k z t1k z ttk z t1k z t1 tk z tttk z ttk z tts

k z tk z tk z t1k z t12222222222

.(11)

After differentiation, we get

ˆt tk t zs st tk t zt tt tk t zk t zk t zt tk t zk t zs ss st tt tt tk t zk t zk t zk t zk t zk t zk t zk t zt tk t zt tk t zt tt t

ˆt t

ˆk t zk t zt tk t zt tˆ

t tˆk t zt tˆk t zk t zk t z

s ss ss sk t zk t z .

(12)

Now, since t tt ts s

t tt tt tt ts ss s

t tt tt tt tt tt tt tt tt tt tt tt tt tt tt tt ts s

t ts s

t t , we calculate the absolute value of the vector and get

2 2

ˆt tk t zs s

2 22 2t tk t zk t zk t zk t zt tt tk t zk t zk t zk t zk t zk t zk t zk t zk t zt tk t zt tk t zt tt tˆ

t tˆk t zk t zt tk t zt tˆ

t tˆk t zt tˆk t zk t zk t zk t z

s ss ss sk t zk t z . Again assuming that the chain is extended, i.e. 1tt 11, since

the tangent fluctuations are small as the extension approaches the length of the molecule, we get

221t tk t

s s

2t t2t tt tt tk tk tk tt tt t2t t2k tk t1k t1k tk t1k t1k tk tk tk t1k t1t tk tt tk tt t2t t2t t2k t1k t1t tk tt t1t t1k t1t t1s ss s

k t1k t1k tk ts ss ss s

k t1k t1k t .(13)

Then, we can insert 2

2 tks

22tttttss

into Eq. (8) to obtain the quadratic approximation [35]

Page 12: Stretching Dna

12

22

0

12

L

elastictE EI Ft ds FLs

22t12

E EI Ft ds FLE EI Ft ds FL1E EI Ft ds FL1E EI Ft ds FLE EI Ft ds FLE EI Ft ds FL2E EI Ft ds FLE EI Ft ds FLE EI Ft ds FL2E EI Ft ds FL22E EI Ft ds FL2E EI Ft ds FL2E EI Ft ds FLE EI Ft ds FLE EI Ft ds FLE EI Ft ds FLE EI Ft ds FLE EI Ft ds FLE EI Ft ds FLs

L tE EI Ft ds FLttE EI Ft ds FLE EI Ft ds FLttE EI Ft ds FLE EI Ft ds FLE EI Ft ds FLE EI Ft ds FLE EI Ft ds FLE EI Ft ds FLE EI Ft ds FLE EI Ft ds FLtE EI Ft ds FLtE EI Ft ds FLtE EI Ft ds FLttE EI Ft ds FLtE EI Ft ds FLttE EI Ft ds FLtE EI Ft ds FLtE EI Ft ds FLE EI Ft ds FLE EI Ft ds FLss

,(14)

where we have used 0

L

zz t dsz t ds0

L

z0

L

zz t dszz t dsz and 0

LL dsL ds

0

L

0

LL ds in

2

0 01

2L L

ztFz F t ds F ds

2tFz F t ds F ds2

L L

0 00 0z0 0

2t0 0

L L

z0 0z0 0Fz F t ds F dszFz F t ds F dsz

ttFz F t ds F dsFz F t ds F ds1Fz F t ds F ds1Fz F t ds F dsFz F t ds F dstFz F t ds F ds1Fz F t ds F ds1 tFz F t ds F dsttFz F t ds F dstFz F t ds F dst222

. Next, we

perform Fourier transformation, defining ( ) ( )iqst q e t s dst q e t s ds( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )iqst q e t s ds( ) ( )t q e t s ds( ) ( )iqst q e t s dsiqs( ) ( )iqs( ) ( )t q e t s ds( ) ( )iqs( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )( ) ( )iqs( ) ( )t q e t s ds( ) ( )iqs( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( )t q e t s ds( ) ( ) , which in Fourier space

yields

221 1 ( )2 2elasticE Aq F t q dq FL1 11 12 2

E Aq F t q dq FLE Aq F t q dq FL1 1E Aq F t q dq FL1 12 2

E Aq F t q dq FL2 21 1E Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FL1 1E Aq F t q dq FL1 1E Aq F t q dq FL1 1E Aq F t q dq FLE Aq F t q dq FL2 22 2

E Aq F t q dq FL2 22 22 2

E Aq F t q dq FL2 2

E Aq F t q dq FL2 22 2

E Aq F t q dq FL2 2

E Aq F t q dq FL2 2

E Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FL2E Aq F t q dq FLE Aq F t q dq FL2E Aq F t q dq FL2E Aq F t q dq FLE Aq F t q dq FL2E Aq F t q dq FL2E Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FL2E Aq F t q dq FL2E Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FL222E Aq F t q dq FLE Aq F t q dq FL2E Aq F t q dq FL2( )E Aq F t q dq FL( )E Aq F t q dq FLE Aq F t q dq FL( )E Aq F t q dq FL( )E Aq F t q dq FLE Aq F t q dq FL( )E Aq F t q dq FL( )E Aq F t q dq FLE Aq F t q dq FL( )E Aq F t q dq FL( )E Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FL( )E Aq F t q dq FL( )E Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FL( )E Aq F t q dq FL( )E Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FLE Aq F t q dq FL ,(15)

where A=EI. In Fourier space, the energy is the sum of the decoupled harmonic degrees of freedom for each Fourier mode. According to the equipartition theorem [5] each normal mode has an energy / 2Bk T where the multiplier 2 accounts for the x- and y-components of tt :

22

2

2

1 2 ,2 2

2 .

B

B

k TAq F t

k TtAq F

2 ,2 ,2 22 22 2

2 ,2 2

2 ,2 ,B2 ,k T2 ,k T2 ,Bk TB2 ,B2 ,k T2 ,B2 ,Aq F tAq F tAq F tAq F t2 2

Aq F t2 22 2

Aq F t2 22 2

Aq F t2 2

2Bk TBk TB

Aq F2Aq F2 .Aq FAq F

2

2 22 2k T

2 22 22 22 2 ,2 ,

2 22 ,

2 22 ,2 ,B2 ,k T2 ,k T2 ,Bk TB2 ,B2 ,k T2 ,B2 ,Aq F t

2 2Aq F t

2 22 22 22 22 2k Tt

Aq F2k T

Aq F(16)

We can now use Parseval‟s theorem [31] to calculate 2 ( )t s( )t s( )t s( )

2 22

2

2

1( ) ( ) ( )2

arctan /2 ,2

q

B B B

t s t q t q dq

Aq FAk T k T k TdqAq F FA FA

2

2 Aq F

t s t q t q dqt s t q t q dq( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )

k T k T k TAq F

t s t q t q dq( ) ( ) ( )t s t q t q dq( ) ( ) ( )t s t q t q dq( ) ( ) ( )t s t q t q dq( ) ( ) ( )

2arctan /2arctan /2Aq FAAq FAAq FAarctan /Aq FAarctan /2arctan /2Aq FA2arctan /2k T k T k Tarctan /k T k T k Tarctan /Aq FAk T k T k TAq FAAq FAk T k T k TAq FAAq FAk T k T k TAq FAarctan /Aq FAarctan /k T k T k Tarctan /Aq FAarctan /k T k T k Tarctan /k T k T k Tarctan /Aq FAk T k T k TAq FAAq FAk T k T k TAq FAAq FAk T k T k TAq FAAq FAk T k T k TAq FAarctan /Aq FAarctan /k T k T k Tarctan /Aq FAarctan /

B B Bk T k T k TB B Bk T k T k TB B B222

B B Bk T k T k Tarctan /k T k T k Tarctan /B B Bk T k T k TB B B

Aq FAk T k T k TAq FAAq FAk T k T k TAq FAAq FAk T k T k TAq FAAq FAk T k T k TAq FAarctan /Aq FAarctan /k T k T k Tarctan /Aq FAarctan /B B B

Aq FAB B Bk T k T k TB B B

Aq FAB B Bk T k T k Tarctan /k T k T k Tarctan /Aq FAk T k T k TAq FAAq FAk T k T k TAq FAAq FAk T k T k TAq FAarctan /Aq FAarctan /k T k T k Tarctan /Aq FAarctan /

FA FAFA FAFA FAFA FA,

FA FAFA FAFA FAFA FA,

Aq F FA FAFA FAFA FAFA FAFA FA

2t s t q t q dqt s t q t q dqt s t q t q dq( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )t s t q t q dqt s t q t q dqt s t q t q dq( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )

k T k T k Tk T k T k TAq F2Aq F2Aq F

B B Bk T k T k TB B Bk T k T k TB B BdqB B BdqB B BB B Bk T k T k TB B BdqB B Bk T k T k TB B B2Aq F2Aq F2Aq F

dqAq F

2 212 212 2

222 2

222 22 212 22 2t s t q t q dq2 22 212 2t s t q t q dq2 212 2( ) ( ) ( )t s t q t q dq( ) ( ) ( )2 2( ) ( ) ( )2 2t s t q t q dq2 2( ) ( ) ( )2 21( ) ( ) ( )1t s t q t q dq1( ) ( ) ( )12 212 2( ) ( ) ( )2 212 2t s t q t q dq2 212 2( ) ( ) ( )2 212 2

2( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )

2t s t q t q dq

2( ) ( ) ( )

2( ) ( ) ( )t s t q t q dq( ) ( ) ( )

2( ) ( ) ( )

2t s t q t q dq( ) ( ) ( )t s t q t q dq( ) ( ) ( )

2t s t q t q dq

2( ) ( ) ( )

2( ) ( ) ( )t s t q t q dq( ) ( ) ( )

2( ) ( ) ( )2 22 2

222 22 2t s t q t q dqt s t q t q dq2 2t s t q t q dq2 22 2t s t q t q dq2 2( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )2 2( ) ( ) ( )2 2t s t q t q dq2 2( ) ( ) ( )2 22 2( ) ( ) ( )2 2t s t q t q dq2 2( ) ( ) ( )2 2

2t s t q t q dqt s t q t q dqt s t q t q dq( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )

2t s t q t q dq

2( ) ( ) ( )

2( ) ( ) ( )t s t q t q dq( ) ( ) ( )

2( ) ( ) ( )2 22 2t s t q t q dqt s t q t q dq2 2t s t q t q dq2 22 22 2t s t q t q dqt s t q t q dqt s t q t q dq2 22 22 22 2t s t q t q dqt s t q t q dq2 2t s t q t q dq2 22 2t s t q t q dq2 22 2t s t q t q dq2 22 2t s t q t q dq2 2( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )2 2( ) ( ) ( )2 2t s t q t q dq2 2( ) ( ) ( )2 22 2( ) ( ) ( )2 2t s t q t q dq2 2( ) ( ) ( )2 22 2( ) ( ) ( )2 2t s t q t q dq2 2( ) ( ) ( )2 2t s t q t q dqt s t q t q dq( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )t s t q t q dqt s t q t q dqt s t q t q dq( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )( ) ( ) ( )t s t q t q dq( ) ( ) ( )

(17)

The extension can be expressed as a ratio of end-to-end distance and contour length:

2 2

0 0

ˆ ˆ1 1ˆ ˆ 1 1 1

2 2 4

L LB

t t k Tz t z ds dsL L L FA

k T2 22 22 2ˆ ˆ2 2ˆ ˆ2 22 2ˆ ˆ2 22 2ˆ ˆ2 2t tt tt t2 2t t2 22 2t t2 22 2t t2 2t tt tt tt tt tt t

1 1 11 1 11 1 14

Bk TBk TB

FA1 1 11 1 1 Bk TBk TB1 1 11 1 11 1 11 1 11 1 1

t tt t1 1L L L 2 22 2

1 1 11 1 12 22 2

L L t tt tt tt tt tt tt t1 11 1L L1 1L L L

t z ds ds1 1 1t z ds dst z ds ds1 1 1t z ds ds1 1 11 1 1t z ds ds1 1 11 1 1t z ds ds1 1 11 1 1t z ds dst z ds ds1 1 1t z ds ds1 1 11 1 1t z ds ds1 1 1t tt tt t1 11 1

ˆ1 1t z ds ds1 1t z ds ds1 11 1

ˆ1 1t z ds ds1 1

ˆ1 1

L L L 2 22 21 1 1

2 2t z ds ds1 1 1t z ds ds1 1 11 1 1t z ds ds1 1 1

2 2

L L1 1L L1 11 1L L1 1L L1 1L L L

1 11 1ˆ

1 1t z ds dst z ds ds1 1t z ds ds1 1ˆt z ds dsˆ

1 1ˆ

1 1t z ds ds1 1ˆ

1 1L L L

.(18)

This shows that z approaches the contour length L of the molecule with a 1/ F behaviour when the force is large [35]. This can be further manipulated to

Page 13: Stretching Dna

13

22

1( )

4 1B

FAk T z

L

12zz4 14 1 z

L4 1

LL

, (19)

where A=EI=kBTP, and P is the persistence length of the molecule. Equation (19) represents a high-force approximation formula of the WLC model. It is valid when the force is large (z/L>0.9).

In practice an interpolation formula that is asymptotically correct at both high and low forces is often used. As z→0, Eq. (19) shows that the nondimensional force FP/kbTapproaches 1/4. However, at low extension the force must approach zero. Therefore the terms z/L-1/4 are added to Eq. (19) to yield an interpolation formula that performs well at both low– and high forces:

21 1

44 1B

FP zk T Lz

L

21 1FP z1 1FP z1 1

k T L 4k T L2k T L2k T Lzk T L2 4k T L2k T L2k T Lk T Lzk T L4 14 1

k T Lk T Lzk T Lzk T LL

4 1LL

. (20)

This often used interpolation formula was derived in 1995 by Marko and Siggia [29]. An improved approximation was proposed by Bouchiat et al. in 1999. They subtracted the interpolation formula from the exact numerical solution of the WLC model [36] and expressed the residuals as a seventh-order polynomial. This improved the accuracy of the fit to <0.01% over the useful extension range, where the error in the original approximation was typically ~5%. The improved solution is given by [36]

7

22

1 14(1 / ) 4

ii

iiB

FP z zk T z L L L

7FP z z7FP z z7

2k T z L L L2k T z L L L2

iFP z zFP z zFP z z1 1FP z z1 1k T z L L L4(1 / ) 4k T z L L L4(1 / ) 44(1 / ) 44(1 / ) 4k T z L L L4(1 / ) 424(1 / ) 42k T z L L L24(1 / ) 4224(1 / ) 4k T z L L L4(1 / ) 424(1 / ) 42k T z L L L24(1 / ) 42k T z L L Lk T z L L LFP z ziFP z zi

k T z L L Lik T z L L LiFP z z7FP z z7

k T z L L L2k T z L L L2

iFP z zFP z zFP z zk T z L L LiFP z zk T z L L Lik T z L L LiFP z zFP z zFP z zFP z zFP z zk T z L L Lk T z L L Lk T z L L Lk T z L L Lk T z L L L

,(21)

where α2 = −0.5164228, α3 = −2.737418, α4 = 16.07497, α5 = −38.87607, α6 = 39.49944, and α7 = −14.17718.

A low-force approximation can now also be derived. Taking the Taylor expansion from the

right hand side of Eq. (19) when z→0 yields 2 3

2 3

1 2 3 41 ...4

z z zL L L

2 31 2 3 42 31 2 3 42 3z z z1 2 3 4z z z1 2 3 41 2 3 4z z z1 2 3 4z z z1 2 3 41 ...1 2 3 41 ...1 2 3 41 ...1 2 3 42 31 ...1 ...1 ...1 ...2 31 ...2 3

1 2 3 41 ...1 2 3 41 ...1 2 3 4z z z1 2 3 4z z z1 2 3 41 ...z z z1 ...1 2 3 41 ...1 2 3 4z z z1 2 3 41 ...1 2 3 4L L L2 3L L L2 3L L L2 32 3L L L2 3L L L2 3 , and omitting

terms of z2 and higher powers of z, we insert the two first terms into Eq. (19). This gives us the Hookean low-force approximation (identical to the FJC low-force approximation Eq. (7))

1 1 3 .4 2 4 2B

FP z z zk T L L LFP z z z1 1 3FP z z z1 1 3FP z z z1 1 3k T L L L4 2 4 2k T L L L4 2 4 2FP z z zFP z z zk T L L Lk T L L L

.k T L L L

(22)

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14

In Fig. 4 the improved WLC interpolation formula (Eq. (21)) and the FJC-model (Eq. (4))

are compared. Furthermore, the WLC/FJC low-force- (Eqs. (7 & 22)) and WLC high-force

approximations (Eq. (19)) are included in the figure to show their contribution to the WLC

interpolation formula.

Figure 4. Theoretical DNA force-extension models compared. Left: the WLC interpolation formula (Eq. (21), blue), the

FJC model (Eq. (3), dashed red), the WLC/FJC low-force approximation (Eq. (7&22), dash-dot magenta), and the WLC

high-force approximation (Eq. (19), dashed green) in linear scale. Middle: the same as in left but the force is in

logarithmic scale. Right: the same as previous, but in logarithmic scale. The plots show the main difference between the

two, namely the intermediate force region (z/L ~0.5-0.9), where the WLC model bends earlier.

2.3 Temperature controller for the objective of the optical tweezers

SM experiments are sensitive and vulnerable to the slightest disturbances. The trapping

laser used in these experiments emits infrared laser light of 1-4 W power. Some of this

light is absorbed by the microscope objective. As a result, the trap is susceptible to thermal

drift when the trap laser is switched on/off or steered. For a linear thermal expansion

coefficient of ~15-6

K-1

for steel, a 0.1 K change in temperature for an objective of ~30 mm

diameter would lead to ~45 nm thermal expansion of its diameter. Similarly, for an

objective of 60 mm parfocal distance the same change in temperature would result in ~90

nm change in length. These result in the laser focus drifting vertically relative to the

surface of measurement chamber, as well as lateral drifting if a sample is trapped away

from the radial center of the objective. During experiments the laser trap is switched off

while searching for beads to trap. When a bead is found, the trap is switched on. This

causes periodic heating of the objective, meaning that to get good results a detector laser

calibration should be performed once in a while. However, this can be avoided by pre-

heating the objective to, for example, 30°C, or slightly above the room temperature. This

ensures that the heat outflux from the objective is much greater than the heat absorbed by

the objective. Thus, the laser will not affect the thermal stability of the objective. To

maintain the desired temperature, a temperature sensing transducer and a heating element

can be attached to the objective. Monitoring the temperature allows use of a feedback loop

to control the heating current in the element.

2.3.1 Temperature measurement circuit

The measurement circuit (see Fig. 5) is a Wheatstone bridge [37] that compares resistance

between two sensing resistors, one of which is a Pt100 [38] platinum temperature

transducer. The Pt100 has temperature dependent resistance and its name derives from its

resistance being 100 Ω at 0°C. When the bridge is in balance, i.e. the voltage difference

between the sensing points (A and B) is 0 V, the Pt100 temperature transducer has the

0 0.2 0.4 0.6 0.8 1

2

4

6

8

10

12

14

16

18

20

Normalized extension (z/L)

No

rmal

ized

fo

rce

(FP

/kB

T)

Linear extension vs. linear force

0 0.2 0.4 0.6 0.8 110

-4

10-2

100

102

104

Normalized extension (z/L)

No

rmal

ized

fo

rce

(FP

/kB

T)

Linear extension vs. log force

10-2

100

10-4

10-2

100

102

104

Normalized extension (z/L)

No

rmal

ized

fo

rce

(FP

/kB

T)

Log extension vs. log force

WLC interpolation formula

FJC model

WLC/FJC low-force approximation

WLC high-force approximation

Page 15: Stretching Dna

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same resistance as the sensing resistor. Therefore, we chose the sensing resistor to be 110 Ω, which is the nearest standard resistor to 111.67 Ω, the resistance of Pt100 at 30°C. We set the current through the bridge to be 1 mA, which is distributed equally in the bridge when the resistances are at balance. The voltage difference is then amplified 2084-fold to ±10 V using an INA111 [39] instrumentation amplifier. A custom-made LabVIEW 8.6 (National Instruments, Austin, TX) program converts the voltage difference into temperature according to a linear calibration.

Figure 5. Schematic of the temperature measuring circuit. The Pt100 in the measuring bridge is three-wire connected to reduce the effect of wire resistance. The RF filter removes the EMI-effect the acousto-optic deflector wires have on the instrumentation amplifiers (DC offset). The output of the INA111 is further low-pass filtered with a 10 Hz active 2nd order Butterworth stage. The signal is read by the analog input of a data acquisition card (PCI-6014, National Instruments, Austin, TX).

The circuit board consists of three parts; the bridge, the amplifier, and the low-pass filter. An external ±15 V power source supplies the power to the circuit. A +5 V voltage reference [40] (REF02, Texas Instruments, Dallas, TX), converts the 15 V supply voltage to 5 V ± 0.2% used in the bridge. At the balance point the current through the Pt100 is ~0.5 mA, sothe resistors R2 were calculated as

2 25111.67 10 9.9

0.5seriesbal

V VR Ω R kΩ R kΩ.I mA2 2111.67 10 9.92 2111.67 10 9.92 2seriesV V5V V5111.67 10 9.9V V111.67 10 9.9RseriesRseriesV VRV V

Ω R kΩ R kΩ.111.67 10 9.9Ω R kΩ R kΩ.111.67 10 9.9111.67 10 9.9Ω R kΩ R kΩ.111.67 10 9.92 2111.67 10 9.92 2Ω R kΩ R kΩ.2 2111.67 10 9.92 2V V

Ω R kΩ R kΩ.V V5V V5

Ω R kΩ R kΩ.5V V5111.67 10 9.9V V111.67 10 9.9Ω R kΩ R kΩ.111.67 10 9.9V V111.67 10 9.95111.67 10 9.95V V5111.67 10 9.95

Ω R kΩ R kΩ.5111.67 10 9.95V V5111.67 10 9.95

I mA2 2I mA2 22 20.52 2I mA2 20.52 2seriesI mAseries(23)

The closest standard resistor is 10 kΩ. Because the sensing resistor is 110 Ω the temperature of the balance point changes to ~25.68°C according to the Pt100 resistance equation

20 (1 ),TR R AT BT 20 (1 ),2(1 ),2R R AT BT0R R AT BT0 (1 ),R R AT BT(1 ), (24)

where RT is the resistance at certain temperature, T the temperature in °C, A is 3.9083×10-3

°C-1, and B is -5.775×10-7 °C-2.

The current over the bridge is

2 10 0.99 .10100parallel series

U U VI mAR R Ω

0.99 .10100

U U V2 10U U V2 10I mAI mAI mAI mA0.99 .I mA0.99 .U U VI mAU U V2 10U U V2 10I mA2 10U U V2 10R R Ω

(25)

At 50°C the resistance of the Pt100 is ~119.4 Ω. This is chosen as the maximum

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temperature, i.e. at this temperature the output of the circuit is at its minimum (Pt100 is in the negative terminal), -10 V. Then the voltage at point A is

119.4 5 59.0 .119.4 10

V mVk

119.4 5119.4 10

119.4 5V 59.0 .mV59.0 .mV59.0 .59.0 .mV59.0 .mV59.0 .59.0 .119.4 10k

(26)

According to the previous equation, at point B the voltage will be, with a 110 Ω sensing resistor, 54.4 mV. The difference between voltages at points A and B (4.6 mV) will be amplified properly to give the minimum -10 V output for the circuit. The gain and gain resistor are calculated as

10 2174,4.6

VGmV

10 2174,4.6

VmV

50 23 ,2173g

kR 50 23 ,2173

k 23 ,23 ,23 ,(27)

where G is the gain and Rg is the gain resistor for the INA111. The closest standard resistor is 24 Ω. From Eq. (27) we get a gain of 2084 which gives 4.8 mV maximum voltage difference at points A and B. Further, at -10 V output voltage, the voltage at point A should be 59.2 mV, which corresponds to ~51.1°C. The output equation for the circuit is

1 1

1 1 2

501 ,B Bout

PT g

RV RV kVR R R R R

RV RV 50kRV RV 50k50k1 1B B1 1B B1 11 1B B1 1B B1 1RV RV1 1RV RV1 1B BRV RVB B1 1B B1 1RV RV1 1B B1 1 1 ,1 ,1 ,1 ,501 ,501 ,50k1 ,k1 ,1 ,1 ,1 ,1 ,1 ,1 ,R R R R R

1 ,R R R R R

1 ,1 1 2PT g1 1 2PT g1 1 2R R R R R1 1 2R R R R R1 1 2R R R R RR R R R R1 1 2PT g1 1 2PT g1 1 2R R R R R1 1 2R R R R R1 1 2PT gR R R R RPT g1 1 2PT g1 1 2R R R R R1 1 2PT g1 1 2R R R R RR R R R R

1 ,R R R R R

1 ,R R R R R

1 ,PT gPT gR R R R RPT gPT gR R R R RPT gR R R R RPT gR R R R RR R R R RR R R R RPT gPT gR R R R RPT gR R R R RPT gR R R R RPT gR R R R RR R R R RR R R R R

(28)

where the first factor is the voltage difference between points A and B in the bridge and the second factor is the gain of the INA111. With a 119.4 Ω Pt100 resistance, corresponding to a temperature of 50°C, the output is -9.48±1.23 V. The error is the expected uncertainty in the design phase; however, once the calibrations are done the true error will be smaller. Wecalculated the errors using the least squares method [41]. The circuit includes four temperature sensing bridges and one REF02. The 20 kΩ resistors in the RF filter do not alter the bridge voltage since virtually no current enters the INA111 inputs. Furthermore, we installed a 2nd order active Butterworth low-pass filters on the INA111 outputs. The filter was dimensioned according to recommendations by the instrumentation amplifier guide provided by Analog Devices [42]. The filters are based on the Sallen-Key [43] topology and they are designed to have cutoff at ~10 Hz with unity amplification. The corresponding transfer function is

2

2

1( )21

C C

H ss s22 s s

2C C

12222 s s

2

. (29)

The temperature measurement circuit schematic and PCB etching mask are shown in Appendix A.1.

2.3.2 Temperature control circuit

The purpose of the temperature control circuit (Fig. 6) is to drive the heating wire with apower up to 6 W. It uses an operational amplifier (TL071 [44]) to control the gate voltage of an n-channel power MOSFET (IRF630 [45]). The op-amp attempts to nullify the difference between its positive and negative inputs by adjusting the output voltage. The

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drain-source current of the MOSFET is directly proportional to the gate voltage. The current does not depend on the resistance of the heating wire nor the temperature- and current-dependent resistance of the drain-source junction (Rds) since the feedback always sets the negative terminal to the same voltage as the positive terminal. This means that the current through the heating-wire is

,1.1

VI1.1

V ,V (30)

where V- is the voltage at negative terminal and 1.1 Ω is the resistance of two parallel 2.2 Ωresistors.

Figure 6. The temperature control circuit. The control voltage is divided from 0-10 V to 0-0.56 V. The op-amp sets the MOSFET’s gate voltage such that the voltage difference between the positive- and negative terminals of the op-amp is zero. The diode protects the MOSFET from current spikes.

The 0-10 V control voltage is divided to give 0-0.56 V at the positive terminal of the op-amp. At maximum input of 10 V the op-amp adjusts the gate voltage of the MOSFET such that the voltage at the negative terminal is 0.56 V. This gives ~0.5 A current through the heating wire. Considering a wire of 19 Ω resistance, the heating wire dissipates ~4.8 W of heat. Between the output of the TL071 and the gate of the IRF630 is a passive low-pass filter that removes possible oscillations from the gate voltage. The filter has cutoff frequency of ~16 Hz. The temperature control circuit schematic and the PCB etching mask are shown in Appendix A.2.

2.3.3 Feed-forward- and feedback controller

We use a feed-forward- and feedback controller to achieve precise temperature control. The feed-forward (FF) is an open-loop controller that uses pre-calibrated linear voltage-temperature curve. This is based on that we roughly know what temperature the heating voltage will produce, even though the calibration depends on room temperature and air currents of the room. The FF is useful to provide constant power to achieve a temperature near the setpoint/target temperature.

The proportional-integral-derivative (PID) feedback controller uses the error between the setpoint and the measured temperature to adjust the control signal. The proportional control calculates the error and multiplies it with a preset gain value. With too large a gain the P-term tends to oscillate. However, should the P-control be stable, it will often contain a steady-state error. The integral term calculates the sum of the previous errors point-by-

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point. The previous errors are recorded into a queue of certain length. The I-term removes the steady-state error from the controller. However, the I-term causes a long settling time, or ringing, and with too large a gain it causes uncontrolled oscillation. The D-term works on the same principle as the integral term. However it calculates the slope between previous errors. Its purpose is to prevent a fast rise or descend of temperature, thus damping and stabilizing the oscillations produced by the P- and I-terms. Too large a gain in the D-term makes the controller react to rising or descending temperatures strongly, which again causes the system to oscillate. The feedback- and the FF signals are summed to produce the control signal. The output equation including PID [46] and feed-forward is given by

0( ) ( ) ( ) ( )

t

out p i d FF setdeV t K e t K e d K V Tdt

( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )deV t K e t K e d K V T( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )deV t K e t K e d K V Tde( ) ( ) ( ) ( )de( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )de( ) ( ) ( ) ( )out p i d FF setdtout p i d FF set( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )dt

( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )de( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )deV t K e t K e d K V Tde( ) ( ) ( ) ( )de( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )de( ) ( ) ( ) ( )out p i d FF setdtout p i d FF set( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )dt

( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )t de0out p i d FF set0out p i d FF set0 dtout p i d FF setdtout p i d FF set

t

out p i d FF set0out p i d FF set0( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )

0( ) ( ) ( ) ( )

0out p i d FF set0( ) ( ) ( ) ( )

0V t K e t K e d K V T( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )deV t K e t K e d K V T( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )deV t K e t K e d K V Tde( ) ( ) ( ) ( )de( ) ( ) ( ) ( )V t K e t K e d K V T( ) ( ) ( ) ( )de( ) ( ) ( ) ( )out p i d FF setdtout p i d FF set( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( )

dt( ) ( ) ( ) ( )out p i d FF set( ) ( ) ( ) ( ) , (31)

where pK , iK , and dK are the proportional-, integral-, and derivative-gains,

( ) set measurede t T Tset measurede t T Tset measurede t T Tset measured is the error between setpoint and the temperature, and ( )FF setV T is the FF voltage. The system diagram is presented in the Fig. 7.

Figure 7. The PID/feed-forward system diagram. Controlling the temperature consists of two terms: the prediction of the heating voltage at setpoint temperature (feed-forward) and the error between the setpoint and the current temperature (processed by PID).

We implemented two sets of PID gains; far from the setpoint only P-gain is used and its only purpose is to make the controller get near the setpoint fast, and closer to the setpoint another set of gain parameters is used when the difference between the temperature and the setpoint is smaller than 0.3°C. These values are designed to settle the temperature around the setpoint. The controller measures and controls the temperature every 0.02 seconds.

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3. Materials and methods

3.1 Preparation of biotin-digoxigenin labelled DNA by PCR

In this experiment we used 10kb dsDNA constructs made with a standard PCR program

(Fig. 8 & 9). The template was phage-λ viral DNA (bacteriophage lambda

(cI857ind 1 Sam 7), New England Biolabs, Ipswich, MA, USA). In the first cycle the PCR

program separates the double-stranded template by heating the solution. When the solution

is cooled down primers anneal to the predefined positions in the now-separated strands of

the template. The oligonucleotides were forward 5‟digoxigenin

GTGGAATGAACAATGGAAGTCAAC 3‟ and reverse 5‟biotin

CGAACACTTTCCCGCAGAAAC 3‟, both manufactured by ThermoScientific (Rockford,

IL). A polymerase (Dynazymes EXT, Finnzymes) then starts to incorporate the bases

complementary to the template from the 3‟ end of the primer onward. As a result the

solution will contain the original templates and similar amounts of the shorter DNA

sequences built from the oligos.

In the second cycle, the solution is again heated so that the new double-stranded

synthesized strands separate. This time there are the original separated strands and the

same number of synthesized strands in the solution. As the solution is again cooled down,

the unattached oligonucleotides anneal to the strands. In this cycle, the polymerase

manufactures strands of desired length.

In the third cycle, after heating, we have the original strands from the template, double

amount of shorter DNA sequences, and the same amount of correct length DNA sequences

as the original strands. The same procedure is performed as before. Hence, this procedure

is repeated several times, causing exponential growth in the amount of synthesized selected

DNA. In the end, the relative amounts of the original template and the byproduct are

negligible compared to the number of strands of the selected DNA sequence.

Figure 8. Schematic of the PCR program. The arrows depict the primers. End products of the selected DNA sequence are

produced in the third cycle. In cycles after that, the number of the end products increases exponentially.

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Figure 9. a) Heating separates the dsDNA strands of the template (red and green) allowing the primers (yellow and

blue) to attach to their complementary sequences. b) The result of the PCR program is a 9942 bp (~3.38 µm) long strand

of BIO-DIG labelled dsDNA.

3.2 Preparation of dumbbell assays

We form DNA dumbbell constructs (see Fig. 10) by attaching the 5‟biotin and the

3‟digoxigenin ends to, respectively, streptavidin- and protein G-coated beads. The DNA

constructs are prepared at Finnish Centre of Excellence in Virus Research (Dennis

Bamford‟s group, University of Helsinki, Institute of Biotechnology), and stored in -20°C

at the Electronics Research Laboratory. The beads are commercially available from various

companies. Streptavidin has high affinity for the vitamin biotin [47], attached to the end of

the DNA strand. They form a strong bond binding the DNA onto the bead. Similarly, the

steroid digoxigenin binds strongly on anti-digoxigenin antibody that can be attached to a

bead coated with protein G [13]. TEW buffer is used for washing the laminar flow

chamber and diluting the sample. It contains Tris (tris(hydroxymethyl)aminomethane),

which acts as a buffer, EDTA (ethylenediaminetetraacetic acid) binds divalent cations in

the solution (to prevent enzymatic activity in the sample), and BSA (Bovine serum

albumin) prevents the DNA and proteins from binding to the glass walls of microfluidic

chamber. NaCl is used to create osmotic conditions similar to those in cells.

TEW (Tris-EDTA-water) and CB (chamber block) buffers contain 20 mM of Tris (pH 8),

1mM of EDTA, 150 mM of NaCl, and 0.01% by volume Polysorb 20 (Tween 20), BSA,

0.05 mg/ml for TEW and 5 mg/ml for CB. These buffers are then filtered through a 0.2 µm

syringe filter. To prevent DNA from adhering onto the channel, the channel is blocked by

injecting 0.5-1 ml of CB-buffer into it. After ~30 min incubation in room temperature, we

wash the channel with same amount of TEW-buffer.

3.2.1 Streptavidin beads

To prepare streptavidin (STR) beads we first dilute 10 µl of 0.43 pM streptavidin beads

(0.97 µm diameter, CP01N, Bangs Laboratories, Fishers, IN, USA) in 1 ml TEW buffer.

Then we centrifuge the solution for 5 min at 13000 rpm. We carefully remove the solution

from the centrifuge tube, leaving only the bead pellet. Then we resuspend the pellet in

50 µl TEW and sonicate the solution with six ultrasonic pulses of one second duration and

20 W power. This procedure separates the beads which may have aggregated and removes

unbound streptavidin. Then we take 10 µl of the solution and mix it with DNA. The

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amount of DNA depends on the required molarity and type of DNA. We found that a 500:1

DNA:STR-bead produced tethers easily. To prepare this sample 10 µl of prepared STR

beads were mixed with 18.5 µl of 41 pM DNA-construct. The DNA-bead solution is then

incubated at room temperature for 15-30 min vortexing gently once in a while.

3.2.2 Protein G beads

Initially, protein G beads (2.1 µm diameter, PC-PG-2.0, G. Kisker GbR, Steinfurt,

Germany) are coated with anti-DIG antibody (mouse monoclonal, Roche, Basel,

Switzerland). The mixture may vary according to desired ratio of protein G and

streptavidin beads. In our experiments we added 5 µl of antibody to 10 µl of PG beads (1.6

pM). After that, the solution was incubated at room temperature and vortexed for ~15 min.

Then the solution was diluted in 1 ml of CB buffer, vortexed, and centrifuged for 10-15

min. The CB was discarded and the beads (a white pellet) were resuspended in 50 µl of

TEW. The solution was sonicated with 5 one second pulses of 20 W power.

3.2.3 The sample mix

The sample mix was prepared by mixing 120 µl of STR+DNA beads with 5 µl of PG+anti

DIG beads. This solution was then finalized by diluting with 1 ml of TEW and vortexed.

Figure 10. Dumbbell construct. The PCR program creates biotin and digoxigenin labeled dsDNA strands. The biotin and

digoxigenin attach to the 5’ ends of the dsDNA covalently (blue arrows). Mixing of the STR beads and the DNA bonds the

biotin-labeled end of the DNA non-covalently to STR-coated bead (left). Assembly of the dumbbell bonds non-covalently

the digoxigenin-labeled end of the DNA to anti-digoxigenin antibody, which has previously bonded to the protein G

coating of a bead (right).

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3.3 Laminar flow chambers

We built laminar flow chambers on 76×26×1 mm3 microscope glass slides (Menzel-Gläser,

Braunschweig, Germany). To attach inlet- and outlet tubing, we drilled holes of 1.6 mm

diameter on the microscope slide using a diamond drill (A&F, La Chau-de-Fonds,

Switzerland). The hole locations are marked using a paper template (see Fig. 11). Then we

pressed the tubing (PEEK, inner diameter 0.25 mm, outer diameter 1/16˝, GE Healthcare

Bio-Sciences AB, Uppsala, Sweden) through the holes. The tubing were glued on the

microscope slide by applying optical adhesive (Norland Optical Adhesive 81, Norland

Products, Cranbury, NJ) and illuminating the glue with a UV-lamp for 30 min. To

conclude the attachment the stub of tubing on the bottom-side of the glass was cut off. We

used a template to cut a 2x50 mm2 channel in a 60×20 mm

2 piece of optically clear

adhesive (two-sided tape, 50 µm thick, 3M, Maplewood, MN). We then attached the

adhesive on the bottom-side of the microscope slide. Finally, we attached a 60×24 mm2

cover glass (0.16-0.19 mm thick, Corning, Corning, NY) on the bottom.

Figure 11. Left: the template of a laminar flow chamber, all measurements in millimeters. Right: a ready-to-use laminar

flow chamber. The completed laminar flow chamber construct includes a microscope slide with tubing on the top, a 50

µm thick piece of adhesive with cut-out microchannels in the middle, and a thin cover glass on the bottom.

3.4 Calibration of OT

During experiments two different calibrations are needed: (1) calibration of the position

sensitive detector is used to convert the detection laser center of intensity into position of

the trapped bead, and (2) calibration of trap stiffness is used to convert the measured bead

position into a force.

3.4.1 Calibration of position sensitive detectors

The detector laser signals provide information about the position of trapped objects. The

beams are modeled as Gaussian fields and the trapped objects are approximated as

Rayleigh scatterers [48]. Since we use beads of two different sizes, the calibration must be

done for both types. Firstly, the detector lasers are fixed over the static trap by removing

filters so that one can see the lasers on the screen. Secondly, a bead is trapped in the

steerable trap, which is then placed in the focus of the detector lasers. Thirdly, the

computer makes the calibration by a raster scanning a bead across the detector area. The

visible area we see in the monitor has been calibrated and mapped with the steerable trap.

This gives us information about the position on the screen in MHz scale (AOD calibration,

25-45 MHz). Calibration of the detectors allows us to reveal the detector positions in MHz

and convert the voltage-position surface to metres with a linear least squares fit.

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We model the surface with an equation given by

1

1 2 1 2, 0

( , ) i jij

i jX V V a V V

, 01 2 1 2X V V a V V1 2 1 2X V V a V V1 2 1 2

1

1 2 1 2, 0

i jij1 2 1 2ij1 2 1 2X V V a V V1 2 1 2X V V a V V1 2 1 2

i jX V V a V Vi j

, 0ij1 2 1 2

i j1 2 1 2ij1 2 1 2X V V a V V1 2 1 2X V V a V V1 2 1 2

i jX V V a V Vi j1 2 1 2ij1 2 1 2X V V a V V1 2 1 2ij1 2 1 2 ,

1

1 2 1 2, 0

( , ) i jij

i jY V V b V V

, 01 2 1 2Y V V b V V1 2 1 2Y V V b V V1 2 1 2

1

1 2 1 2, 0

i jij1 2 1 2ij1 2 1 2Y V V b V V1 2 1 2Y V V b V V1 2 1 2

i jY V V b V Vi j

, 0ij1 2 1 2

i j1 2 1 2ij1 2 1 2Y V V b V V1 2 1 2Y V V b V V1 2 1 2

i jY V V b V Vi j1 2 1 2ij1 2 1 2Y V V b V V1 2 1 2ij1 2 1 2 ,

(32)

where X and Y are position coordinates, 1V and 2V the detector voltages, and ija and ijb are the fit parameters. This equation is similar to one proposed by Lang et al. [24], although we use a first-order fit while they used a fifth-order fit. A custom-written Matlab code is used to analyze the data and build the voltage-distance conversion, an example of which is shown in Fig.(12).

Figure 12. Voltage signals given by the raster scan. The x- and y-axes are the positions of the bead and the z-axis is the voltage of the position detector.

3.4.2 Calibration of trap stiffness

We use the power spectrum method [49] to calibrate the stiffness of our trap. The trap is approximated as a Hookean spring with a spring constant k (see Fig. 13). A bead is constantly under Brownian motion. The Langevin equation, neglecting bead mass,

( ) ( ) ( )x t kx t F tx t kx t F t( ) ( ) ( )( ) ( ) ( )x t kx t F t( ) ( ) ( )( ) ( ) ( )x t kx t F t( ) ( ) ( )x t kx t F t( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )x t kx t F t( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )x t kx t F t( ) ( ) ( ) , (33)

describes Brownian motion in a harmonic potential, where F(t) is Brownian thermal force and 6 a6 a is the friction factor. The power spectrum of the Brownian thermal force is given by a white-noise term

2( ) 4noise BS F f k Tnoise BS F f k Tnoise BS F f k Tnoise Bnoise Bnoise BS F f k TS F f k Tnoise BS F f k Tnoise Bnoise BS F f k Tnoise B

2

noise Bnoise B

2

noise Bnoise BS F f k T( ) 4S F f k T( ) 4( ) 4S F f k T( ) 4noise BS F f k Tnoise Bnoise BS F f k Tnoise B( ) 4noise B( ) 4S F f k T( ) 4noise B( ) 4( ) 4noise B( ) 4S F f k T( ) 4noise B( ) 4 , (34)

-1-0.5

00.5

1

-1-0.5

00.5

1-10

-5

0

5

10

X( m)

det 2, XY vs Vx

Y( m)

Vol

tage

X(V

)

-1-0.5

00.5

1

-1-0.5

00.5

1-10

-5

0

5

10

X( m)

det 2, XY vs Vy

Y( m)

Vol

tage

Y(V

)

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where )()(~ 2 tFedtfF ftidt ft is the Fourier transform of ( )F t , and the power spectral

density of a trapped bead is given by

22 2

4( )(2 )

Bbead

k TS x fk f2 22 2

4 Bk TBk TBS x fS x fk f2 2k f2 2k f2 2k f2 2(2 )k f(2 )2 2(2 )2 2k f2 2(2 )2 22 2(2 )2 2(2 )2 2k f(2 )k f(2 )2 2(2 )2 2k f2 2(2 )2 22 22 2k f2 22 2(2 )2 2k f2 2(2 )2 22 22 2(2 )2 2k f2 2(2 )2 2k f

2( )S x f( )S x f( )k f

. (35)

Using Einstein‟s equation for the diffusion constant Bk TD Bk TBk TB and defining the corner

frequency as 1(2 )cf k 1(2 )f k(2 ) we get

2 2 2( )( )bead

c

DS ff f2 2 2( )2 2 2( )2 2 2

cf f( )f f( )2 2 2( )2 2 2f f2 2 2( )2 2 2cf fc( )c( )f f( )c( )D

2 2 22 2 2( )2 2 22 2 2( )2 2 2f f2 2 2( )2 2 22 2 2( )2 2 2( )2 2 2f f( )f f( )2 2 2( )2 2 2f f2 2 2( )2 2 22 2 2f f( )f f( )2 2 2( )2 2 2f f2 2 2( )2 2 22 2 22 2 2( )2 2 2f f2 2 2( )2 2 2 .(36)

Fitting this equation to experimental calibration data gives the corner frequency, from which the stiffness of the trap can be solved. We obtain the data by trapping a bead and measuring the Brownian motion for e.g. 10 seconds. We do this for both traps and both bead types. The data is Fourier-transformed and analyzed to give the stiffness of the trap. Generally, one should decide which type of bead to use with the static- and which to use with the steerable trap (e.g only use smaller ones with the static trap).

Figure 13. The OT potential can be divided into linear- and nonlinear regimes. However, we model the trap as an infinitely linear curve with a stiffness that is the slope of the linear region. If, during experiment, a tether is strong enough so that the OT cannot break it, a bead will be pulled out of the trap.

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3.5 Stretching DNA

Before we do any experiments, we calibrate the detector signal and the trap stiffness. We

then search for suitable samples inside the flow channel. In these experiments, we

arbitrarily chose to trap the smaller STR beads, containing the DNA, in the stronger static

trap, and the larger PG beads in the weaker steerable trap. We bring the beads close to each

other and give ~half a minute to allow the DNA to attach to the PG bead. To get the

statistics for each batch of DNA, we record each bead pair regardless whether it contains a

DNA dumbbell or not. If we find a DNA dumbbell, the computer performs a force-

extension experiment in small steps. A schematic of a dual-trap DNA dumbbell viewed

from the side is shown in Fig. 14.

Figure 14. A strand of dsDNA (green) attached between two polystyrene beads (grey) via antibodies. The detector lasers

(turquoise) measure the position of the beads. The bead on the left is trapped in the stationary trap (red) while the bead

on the right is in the steerable trap (red).

3.6 Building the temperature controller

Building the temperature controller included attaching the Pt100 and the heating wire on

the objective (sec 3.6.1), manufacturing electronics (sec 3.6.2), tuning the feed-forward-

feedback loop (sec 3.6.3), and preparing a LabVIEW software (sec 3.6.4).

3.6.1 Attaching the Pt100 transducer and the heating-wire to the objective

We attached the Pt100 transducer on the top of the objective, onto an inlay next to the lens,

with super glue (Super Attak, Loctite, Düsseldorf, Germany). Should the transducer need

to be removed from the objective, the glue can be softened with acetone. Care had to be

taken when handling the Pt100, since it has two fragile leads of 0.25 mm diameter (Fig.

15).

Figure 15. The Pt100 temperature transducer. The leads have a diameter of 0.25 mm and 15 mm length. The body

dimensions are 2 mm x 5 mm x 0.25 mm. Its temperature range is -70 to 400°C, with ±0.1°C repeatability and the same

amount of drift per year [38].

We wrapped the heating wire (Driver-Harris Co., Harrison, NJ, USA, Ø=0.4 mm, 7 Ω/m,

material unknown) around the objective, from mid-height up to the top. Furthermore, we

installed a piece of double-sided thermally conductive tape (TCDT1, Thorlabs, Newton,

NJ, USA) under the heating wire at the top and bottom of the wrapping to ensure the

fastening onto the objective. We grouped the wire ends of the Pt100 and the heating wire

with a cable tie and soldered them into a four-pin connector (Fig. 16).

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Figure 16. The Pt100 transducer was installed on the top of the objective (Nikon 100x TIRF, 1.49 NA), with heating wire wrapped around the body of the objective. The wire ends of the heating wire and Pt100 were soldered into a four-pin connector.

We calibrated the Pt100 transducers using a dry bath apparatus (AccuBlock, LabnetInternational, Woodbridge, NJ) by filling a heater-well half full with water and placing a 2 ml tube filled with vegetable oil into the bath. Next, we placed the Pt100 transducer and a reference thermocouple into the tube. We heated the bath to five different temperatures, letting the temperature settle for 15 min before recording the output of the temperature measuring circuit. The reference transducer was a Fluke 52 II temperature meter with a chromel-alumel thermocouple. The calibration of channel 1 of the temperature measurement circuit is presented in Fig. 17.

Figure 17. Calibration of Pt100-based temperature measurement circuit. The errorbars are given as the accuracy of the Fluke 52 II, which is 0.05% of the temperature +0.3°C. The accuracy of the Fluke 52 II is also the absolute accuracy of the temperature measurement circuit. The 68% confidence lines for the fit were calculated using the least squares method [41].

-10 -8 -6 -4 -2 020

25

30

35

40

45

50Pt100 calibration

Output voltage (V)

Tem

pera

ture

(Te

mpe

ratu

re (

C)

T3=(-2.58 0.01 C/V)*V+(25.13 0.08) C

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3.6.2 Manufacturing of temperature-measurement and heater circuits

We first draw the etching masks for the temperature measurement- and -control circuits using PADS 2007 circuit design software (Mentor Graphics, Wilsonville, OR). We attached the masks on one-sided circuit board laminates and illuminated them with ultraviolet light (Ultramat II, Bellæmatic, Denmark) to weaken the photoresist over the areas designed to contain no copper. A bath in ~0.7% NaOH solution stripped away the weakened photoresist. We removed the boards from the bath when an image of the circuit appeared on the board. Then we placed the boards into an acid bath (3/5 water, 1/5 HCL, 1/5 H2O2), which etched the excess copper away from around the wires. To finish the circuit boards we drilled the holes, coated the boards with varnish (Plastik 70, CRC Industries, Warminster, PA), and finally soldered the components. See Appendix A for logic schematics and etching masks.

3.6.3 Tuning the feed-forward feedback loop

Before we tuned the PID we had to calibrate the FF. This we did by observing the temperature of the objective while increasing the heating voltage step by step (1 V) during ~30 min. Fitting a line between the points gave the coarse voltage-temperature calibration for the FF. Then we tuned the PID feedback controller by first setting a high P-term and a new setpoint, making the controller oscillate. The P-term was then tuned down a bit. Then we let the system cool down and introduced the D-term. We experimented with different values to provide necessary damping for the system. The system still oscillated somewhat (~3 mKrms), but the mean value was slightly under or over the setpoint. Then we introduced the I-term. This slightly increased the oscillations (~5 mKrms), however they were more accurately around the setpoint. An example of the effect of different control schemes is shown in Fig. 18.

Figure 18. The figure shows how the PID controller was tuned. Firstly, the P-term was introduced and all other gains were set to zero. The proportional controller is shown with a blue line. After introducing the D-term, the oscillations got damped. PD-controller is shown with the red line. Lastly, the I-term did not change the controller much. Inset figure: a close-up of the settled controller at ±10 mK around the setpoint. The red and yellow lines are, respectively, raw data for the PID- and the PD-controllers. The thinner blue- and green lines are the 8th order Butterworth-filtered raw data. The filtering is performed in software.

0 5 10 15 20 25 30 35 40 45 5029

29.5

30

30.5

31

31.5

Time(s)

Tem

pera

ture

(Te

mpe

ratu

re(C

)

Tuning the PID feedback controller

P-controlPD-controlPID-controlSetpoint

40 45 50 5530.99

30.995

31

31.005

31.01

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3.6.4 LabVIEW 8.6 program and DAQ-card with temperature control

The control program was coded with LabVIEW 8.6. We set it to read and write data at 50

Hz frequency. Furthermore, it includes an 8th

order Butterworth filter to remove noise from

the temperature data. Firstly, the program converts the temperature from voltage to

temperature according to a linear calibration (Fig. 17). Secondly, it filters the data using an

8th

order Butterworth low-pass filter. Thirdly, it is processed with the PID/Feedforward

algorithm (see sec. (2.3.3)) to produce the output for the temperature control circuit.

Furthermore, the program includes an optional stepping- and data saving algorithms. The

program schematics are shown in Appendix B.1 and B.2.

The control program receives its input from and feeds its output to a 16-bit PCI-6014 data

acquisition card (National Instruments, Austin, TX). The input range from the temperature

measuring circuit is ±10 V, which means that one digit corresponds to ~0.31 mV. For

channel 1 of the temperature measurement circuit, this correponds to a 0.79 mK

temperature resolution. The output is set by software to give control voltages between 0

and 10 V.

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4. Results 4.1 Contour- and persistence lengths of 10kb DNA

We made SM experiments on two days. During both days one batch of DNA sample was made and used for the whole day. During the first day (Fig. 19) we trapped with 1 and 3 W trap laser power since we had problems with the smaller beads escaping the trap at high DNA-extensions. With 1 W power the static trap exhibited 37.4±0.7 pN/µm stiffness and with 3 W 86.5±3.2 pN/µm stiffness. Respectively, the steerable trap exhibited 20.4±1.0pN/µm and 38.6±0.7 pN/µm stiffness.

Figure 19. Force-extension curve for 10kb DNA (Day 1). Equation (22) was fitted into the force-extension data (thin red lines). On the left, the DNA lengths are comparable to the expected 3.38 µm. However, the persistence lengths do not correspond to the expected ~50 nm. On the right the trap power has been changed to 3 W, which requires re-calibration. Three out of seven samples were inside 10% bounds of 3.38 µm length. Low persistence lengths suggest multiple DNA tethers between the beads. During these experiments ~27% (11/41) of the samples (bead pairs) contained DNA. The errors are estimated with nonlinear least squares method providing 68% confidence intervals.

Increasing the power did not prevent smaller beads from escaping the trap. However, after that the fit gave too long DNA lengths. The turquoise data on the right in Fig. 19 is an example of a dumbbell construct where two or more DNA tethers are attached between the beads. The force starts to increase more at low extension and some DNA tethers may be cut, which is seen as a jumps to lower force at ~3 pN. Also, the lower persistence lengths may be caused by multiple DNA tethers attached on the beads, which results in a gentler slope when the WLC-model starts to bend up.

On the second day (see Fig. 20) we first had 3 W and later 4 W power in the trap. Again, increasing the power did not prevent smaller beads from escaping the trap. This time there was DNA of different lengths, which is a result of unspecific binding of the primers to the template during PCR. Furthermore, some mistakes were also done by starting the computer-driven DNA stretching with beads too far away from each other. This may be seen as data that starts at an extended position with some force already exerted upon it. The consequence is that it makes it more difficult to make a fit into the data. The trap stiffnesses were 18.9±10 pN/µm and 58.9±1.3 pN/µm for the static and 34.3±1.2 pN/µm and 53.3±1.6 pN/µm for the steerable trap, with 3 and 4 W powers respectively. The statistics of the experiments are shown in Fig. 21.

0.5 1 1.5 2 2.5 3 3.5

0

1

2

3

4

5

6

Extension ( m)

Forc

e (p

N)

10kb DNA stretching, 1 W trap power

L=3.413 0.002 m, Lp=23 1 nm

L=3.422 0.002 m, Lp=24 1 nm

L=3.575 0.002 m, Lp=17 1 nm

0.5 1 1.5 2 2.5 3 3.5

0

1

2

3

4

5

6

Extension ( m)

Forc

e (p

N)

10kb DNA stretching, 3 W trap power

L=3.777 0.002 m, Lp=48 1 nm

L=3.890 0.003 m, Lp=71 1 nm

L=3.797 0.003 m, Lp=29 1 nm

L=3.644 0.002 m, Lp=75 1 nm

L=3.932 0.002 m, Lp=15 1 nm

L=3.653 0.003 m, Lp=60 1 nm

L=3.673 0.002 m, Lp=24 1 nm

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Figure 20. Force-extension experiment with 10kb DNA (Day 2). Left: four successful stretching events, neither of which are close to the expected 3.38 μm contour length. The yellow and turquoise plots however are close to the expected persistence length of ~50 nm. Right: five plots that have roughly the same contour length and one that is shorter. In this experiment ~76% (13/17) of the samples contained DNA tethers.

Figure 21. Left: the bin centers are at intervals of 10% of the expected 3.38 µm DNA length (black line) and the bin widths are 10%. DNA tethers were found in 24 samples (out of 58 total), 17 of which were within ±15% of the expected length (bins centered at 3.04, 3.38, and 3.72 µm.). Right: the bin widths are 10 nm. The persistence lengths were not as expected, since only 6 samples had persistence length between 30 and 60 nm. The persistence lengths lower than 30 nm are due to multiple DNA tethers between the beads. The expected persistence length in 150 mM NaCl was ~ 50 nm (black line) [32].

0.5 1 1.5 2 2.5 3 3.5

0

1

2

3

4

5

6

Extension ( m)

Forc

e (p

N)

10kb DNA stretching, 3 W trap power

L=3.153 0.004 m, Lp=36 1 nm

L=3.171 0.006 m, Lp=53 1 nm

L=3.645 0.003 m, Lp=74 1 nm

L=3.630 0.004 m, Lp=63 2 nm

0.5 1 1.5 2 2.5 3

0

1

2

3

4

5

6

Extension ( m)

Forc

e (p

N)

10kb DNA stretching, 4 W trap power

L=2.169 0.002 m, Lp=21 1 nm

L=3.125 0.003 m, Lp=35 1 nm

L=3.221 0.007 m, Lp=15 1 nm

L=3.000 0.003 m, Lp=24 1 nm

L=3.127 0.005 m, Lp=13 1 nm

L=3.233 0.002 m, Lp=38 1 nm

2.37 2.71 3.04 3.38 3.72 4.06 4.390

1

2

3

4

5

6

7

8

DNA lengths ( m)

Coun

t

Distribution of DNA lengths

15 25 35 45 55 65 750

1

2

3

4

5

6

Persistence lengths (nm)

Coun

t

Distribution of persistence lengths

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4.2 Testing the temperature controller We did several experiments to validate and determine the specifications for the temperature controller; the time constants of the Pt100 was measured (sec. 4.2.1), the cut-off frequency of the 2nd order Sallen-Key low pass filter was determined (sec. 4.2.2), a step test was performed (sec. 4.2.3), the long-term closed-loop- and open-loop stabilities were measured (sec. 4.2.4, 4.2.5), the power spectral density of the noise in temperature measurement was estimated (sec. 4.2.6), and finally the stability of the optical trap with open- and closed-loop temperature control was investigated (sec.4.2.7).

4.2.1 The time constants of the Pt100

We measured the rise and fall times for the Pt100 transducer with „cold‟ and „hot‟ objects

(Fig. 22). The „hot‟ object was a tube filled with vegetable oil heated in a dry-bath apparatus (see sec. 3.6.1) to ~38°C and the „cold‟ object was a test microscope objective at room temperature (~24°C). To measure the rise time, we stuck the transducer at room temperature into the heated vegetable oil. Conversely, we measured the fall time by taking the transducer from the oil and pressing it tightly on the test objective. Fitting a sum of exponentials yielded rise time constant 0.48±0.01 s and fall time constant 0.39±0.01 s. The fits were calculated with nlinfit- and the errors for the time constants with nlparci functions of Matlab 7.6.0 [50].

Figure 22. A sum of two exponentials were fit to recorded data of Pt100 response during a fast change in temperature. The exponentials represent a fast component (Pt100) and a slow component (a slow temperature change of the heat bath). The response depends on the heat conduction coefficients of the heat bath.

4.2.2 The cutoff frequency of the 2nd order Sallen-Key low-pass filter

We measured the cutoff frequency of the 2nd order active Sallen-Key low-pass filter (see Sec.2.3.1) using a custom-written LabVIEW program, an oscilloscope (Waverunner 104Xi, Lecroy, Chestnut Ridge, NY, USA), and a signal generator (3314a, Hewlett-Packard, Palo Alto, CA, USA). The oscilloscope was connected to a computer via a crossover ethernet cable and the signal generator via the GBIP-bus. We designed the LabVIEW 8.6 program to control the oscilloscope and the signal generator. The idea is to input a signal from the signal generator to the Sallen-Key filter and simultaneously read its

0 0.5 1 1.5 2 2.5 3 3.5 424

26

28

30

32

34

36

38

Time(s)

Tem

pera

ture

(Te

mpe

ratu

re(C

)

Fitting exponentials on the rise- and fall time datas

Rising temperatureFalling temperatureFit for the rising temperatureFit for the falling temperature

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output with the oscilloscope. As input parameters the program takes the desired

measurement bandwidth (0.1-100 Hz), the number of measurement points (50), and the

desired peak-to-peak modulation amplitude (2 V). The program then instructs the signal

generator to stepwise modulate input signal with the desired frequencies. At the same time

the oscilloscope reads the filter‟s output. The program writes the frequencies and the

corresponding output peak-to-peak amplitude and phase difference relative to the driving

signal into a file on the hard drive. The program diagram of the LabVIEW program is

presented in Appendix B.3.

The program adjusts the seconds/div and the volts/div settings of the oscilloscope to

measure the peak-to-peak amplitude as accurately as possible. The setup works as

spectrum analyzer. The vertical sensitivities are 2 mV – 10 V/div (1 MΩ input resistance)

[51] giving ~74 dB dynamic range. The used frequency range was 0.1 Hz – 1 Hz. As a

comparison, the HP ESA-L1500A spectrum analyzer in our laboratory has a frequency

range of 9 kHz to 1.5 GHz with 78 dB dynamic range [52]. The frequency band of the filter

is presented in Fig. 23.

Figure 23. Measured bandwidth of the 2nd order Butterworth low-pass filter. Fitting the transfer function (see Eq. (31))

yielded a ~5 Hz cutoff frequency.

100

101

-25

-20

-15

-10

-5

0

52nd order Butterworth low-pass filter frequency band

Frequency(Hz)

Am

pli

tud

e(d

B)

Measurement points

2nd order low-pass fit

-3 dB line

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4.2.3 Step tests on the temperature controller

To show that our controller follows the setpoint we used a custom-made LabVIEW 8.6 program (see Appendix B.2) to provide a step input to the temperature setpoint. We made the step test with 125 mK step height and 80 s step duration. The program performed these steps between 31°C and 32°C. The step test results are shown in Fig. 24. It was performed with the objective attached to the optical tweezers setup. The test shows that the temperature follows the setpoint.

Figure 24. Temperature steps of 125 mK were tested on the objective. The green line is the setpoint, driven with stepping algorithm that remains at one setpoint for 80 s. The raw data (red) is Butterworth filtered (blue) in software to improve the overall noise level of the signal.

200 400 600 800 1000 1200 1400

31

31.2

31.4

31.6

31.8

32

Time(s)

Tem

pera

ture

(Te

mpe

ratu

re(C

)125 mK step test

Raw temperature signalFiltered temperature signalSetpoint

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4.2.4 Long-term closed-loop stability

We measured the closed-loop stability for 25 min using Ch1 of the temperature measuring circuit. The results are shown in Fig. 25. We did not use the optical tweezers during this measurement. In the raw data one can see points separated at one digit intervals (~0.79 mK). Fitting a gaussian to the residuals gave a 1.58 mK standard deviation. We calculated the residuals as the deviation of filtered data from the setpoint. Since temperature fluctuations hardly occur at ~50 Hz frequency, we assume the filtered signal to represent the temperature better than the raw signal that is afflicted by electric noise.

Figure 25. Left: we measured the closed loop (PID-loop on) stability of our controller. The temperature was set at 30°C. In the raw data one can see the effect of digitization as the data points are placed at one digit intervals. Right: we calculated the residuals from the Butterworth-filtered data and plotted in a normalized histogram with σ=1.58±0.02 mK

and µ=-0.31±0.03 mK.

0 500 1000 150029.98

29.985

29.99

29.995

30

30.005

30.01

30.015

30.02~25 min closed-loop stability

Time(s)

Tem

pera

ture

(Te

mpe

ratu

re(C

)

Raw dataFiltered datasetpoint

0 0.05 0.1 0.15 0.2 0.25 0.3-8

-6

-4

-2

0

2

4

6

8Distribution of residuals

Normalized count

Tem

pera

ture

resi

dual

(mK

)

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4.2.5 Long-term open-loop stability

We also measured the open-loop stability of our controller, both with and without simultaneous optical tweezers experiment. In Fig. 26 is shown a measurement in open-loop with a simultaneous experiment with optical tweezers.

Figure 26. This figure shows how laser activity causes severe fluctuations in objective temperature. The laser power was 2 W. During this one-hour experiment the temperature rose ~0.5°C. As a comparison, open-loop temperature was stable in a steady-state situation when no trap laser was used and the temperature in the room had stabilized. A full working day had risen the room temperature compared to temperature during OT experiment, which was recorded earlier during day.

4.2.6 Power spectral density

We calculated the power spectral density (PSD) of the closed-loop and open-loop temperature data (Fig. 27). These were compared to the PSD of “dark noise” that was

measured with a 50 Ω resistor in the input of the DAQ card. First, the mean was subtracted from each data, and then the PSD was calculated with the Welch method [53]. This shows that the temperature controller reduces the temperature fluctuations that occur at low frequencies (1 - 100 mHz).

Figure 27. The closed-loop control reduces the slow temperature variations, which is seen as lower PSD at low frequencies (<0.1 Hz) compared to the open-loop control.

500 1000 1500 2000 2500 3000 350023.4

23.5

23.6

23.7

23.8

23.9

24Open-loop stability

Time (s)

Tem

pera

ture

dur

ing

OT

expe

rimen

t (Te

mpe

ratu

re d

urin

g O

T ex

perim

ent (C

)

500 1000 1500 2000 2500 3000 350024.7

24.8

24.9

25

25.1

25.2

25.3

Stea

dy-s

tate

tem

pera

ture

Steady-state temperatureTemperature during OT experiment

10-3 10-2 10-1 100 10110-9

10-8

10-7

10-6

10-5

10-4 Power spectral density

Frequency(Hz)

Pow

er/fr

eque

ncy(

dB/H

z)

Closed-loopDark noiseOpen-loop

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4.2.7 Optical tweezers trap stability

To conclude our experiments with the temperature controller we measured the trap stability of the optical tweezers instrument. We did this both with the temperature control in open- and closed-loop. Firstly, we performed 10 nm steps with a 1 nm trapped polystyrene bead to provide calibration data. Then the bead was kept still for ~5 minutes. The results, seen in Fig. 28, show that with temperature control in open-loop the trap drifted ~1.4 nm/min, compared to ~0.6 nm/min in closed-loop. First we performed the experiment with closed-loop control set at 33°C after which we repeated it with open-loop control set at 7 V(~31°C as feed-forward). We recorded the temperature during the open-loop control experiment, during which it decreased ~0.4°C, and it is included in Fig. 28.

Figure 28. The trap position of the optical tweezers instrument was measured with temperature control in closed-loop and open-loop. In closed-loop (blue) the trap drifted ~0.6 nm/min and the temperature was kept at 33.0°C ± 0.3°mC, while in open-loop (red) the trap drifted ~1.4 nm/min and the controller was set at 7 V (~31°C as feed-forward). During open-loop experiment, the temperature (green) was monitored and it dropped ~0.4°C.

0 50 100 150 200 250 300-50

-40

-30

-20

-10

0

10

20

30

40

50Trap position with temperature control on/off

Time (s)

Trap

pos

ition

(nm

)

0 50 100 150 200 250 30032

32.1

32.2

32.3

32.4

32.5

32.6

32.7

32.8

32.9

33

Tem

pera

ture

(Te

mpe

ratu

re (

C)

Trap position, temperature control onTrap position, temperature control offTrap drifts ~0.6 nm/minTrap drifts ~1.4 nm/minDecreasing temperature, control off

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5. Discussion

First, we performed two single molecule stretching experiments with 10kb λ-phage DNA

to validate the calibration and SM capability of our OT instrument. The correct contour

length for the 10kb DNA is ~3.38 µm. In these experiments the DNA lengths varied from 3

µm to 3.9 µm, excluding the clearly too short tethers. The persistence lengths varied from

15 to 75 nm. The accepted value for the persistence length is 40-60 nm [36]. The values

<30 nm may be credited to multiple DNA tethers between a bead pair, however, the

reasons for persistence lengths >60 nm are unclear. Coarse errors in performing the

experiments are not excluded. The stretching should be started when the beads are next to

each other. To reduce the effect of multiple tethers, smaller DNA concentrations should be

used, although this reduces the likelihood of finding DNA tethers. Also, the escaping beads

proved to be a problem. The cause may be a bad quality of the 0.97 µm beads since there

seems to be deformations in the spherical form, wrongly tuned focus of the trap, or an

objective that absorbs too much of the 1064 nm laser. Furthermore, we have checked the

stiffness calibration with two different methods, one of which is the power spectrum fit,

and the other is the equipartition theorem [54]. Both of the methods give different values

for the stiffness.

Second, we built and tested a precise temperature control device that is capable of

controlling the trapping objective temperature at 1.58 mK precision and 0.3 K absolute

accuracy. Experiments on the spatial position of a trapped 1 µm-diameter bead revealed

that with temperature control in closed-loop (set on 33°C) the amount of trap drift was

halved (~0.6 nm/min) compared to open-loop control (~1.4 nm/min). Since the remaining

part of the drifting may be caused by air currents on the optical table, we recommend

sealing it inside a helium atmosphere. However, the effect of the temperature controller on

experiments with DNA remains to be seen.

6. Conclusions

We conducted a single-molecule DNA stretching experiment to validate the calibration and

SM capability of our OT instrument. We found beads of ~1 µm diameter to often escape

the optical trap. Furthermore, we suggest revising the method we use to calibrate the trap

stiffness. Otherwise, the instrument worked as intended. Furthermore, we improved the

instrument with a temperature controller capable of 1.58 mK precision and 0.3 K absolute

accuracy. The controller reduced the temperature fluctuations in the objective of the optical

tweezers. More importantly, the controller reduced drifting of the optical trap by a factor of

two.

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Appendix

A. Electric schematics

A.1 Temperature measurement circuit

Appendix figure 1. Complete logic schematic of the temperature measurement circuit, drawn using PADS Logic 2007.3

(Mentor Graphics, Wilsonville, OR).

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Appendix figure 2. The solderside etching mask of the temperature measurement circuit (160x100 mm), drawn using

PADS Layout 2007.3 (Mentor Graphics, Wilsonville, OR).

A.2 Temperature control circuit

Appendix figure 3. Logic schematic of the temperature measurement circuit drawn using PADS Logic 2007.3. The logic

schematic does not include the low-pass filter (see Fig. 6) between the output of the TL071 and the gate of the IRF630.

This is because the filter was modded into the circuit after the circuit had already been built.

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Appendix figure 4. The solderside etching mask of the temperature control circuit (102x70 mm), drawn using PADS

Layout 2007.3 (Mentor Graphics, Wilsonville, OR). The etching mask does not include the low-pass filter (see Fig. 6)

between the output of the TL071 and the gate of the IRF630. This is because the filter was modded into the circuit after

the circuit had already been built.

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B. LabVIEW 8.6 programs

B.1 Temperature control program

Appendix figure 5. The temperature control program, designed for LabVIEW 8.6.

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B.2 Step generator program

Appendix figure 6. The step generator for the temperature control program, designed for LabVIEW 8.6.

B.3 Frequency band measurement program

Appendix figure 7. The program diagram of the frequency band measuring LabVIEW program.