Characterization of the sample cell surface

The surface quality of our sample cell surfaces on which the DNA molecules are end-grafted were quantified with AFM¹⁸. The results are presented in Fig.

6.9 where on (a) is a clean glass surface silanized with APTES. The root mean squared roughness was determined to be∼0.5 nm and this result verifies that the coverslips have a very good surface quality. The black spots seen on the surface are holes which have their origin probably in the acid cleaning. In (b) the glass surface is covered with sulfo-SIAB molecules. The surface roughness was almost the same as with silanized glass so we assume that we have indeed a monolayer of sulfo-SIAB molecules. This is also backed up by the periodic structure that is puctually visible. We also found that sulfo-SIAB forms aggregates (data not shown) and therefore filtering the solutions is always recommended.

18Nanoscope III, Digital Instruments, Santa Barbara, CA / US.

(a) (b)

(c)

0 0.2 0.4

Relaxation time [s]

Amplitude

0 0.2 0.4

0 0.2 0.4

0 0.2 0.4

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

0 0.2 0.4

35°

45°

55°

65°

75°

(d)

Figure 6.7: a) EWDLS measurement on thiol-modified end-grafted 2kb DNA carpet cross-linked over sulfo-SIAB to the surface. The measurements were done at angles θ = 35^◦, 45^◦, 55^◦, 65^◦ and 75^◦ with maximum penetration depth of

∼600nm and in V-V geometry. b)Close up of (a): the scattering signal from the surface treated only with sulfo-SIAB at angles35^◦ and 45^◦ shows no measurable correlation function. c) Normalized correlation functions. d) The relaxation time spectra for the normalized correlation functions obtained from CONTIN.

The relaxation time spectrum does not move to faster times as scattering angle is increased therefore we assume that DNA has probably adhered to the surface and we were actually measuring the variations in the static scattered intensity.

(a) (b) (c)

Figure 6.8: a)Sample cell for the EWDLS experiment: the surface of the sample cell was divided with two component silicone rubber into to compartments. The upper compartment was functionalized with sulfo-SIAB and thiol-modified 2 kb DNA whereas the lower compartment had only sulfo-SIAB functionalization. b) The upper surface shows a strong surface scattering signal whereas in c) there is only very weak signal to see.

(a) (b)

Figure 6.9: a) Clean glass surface silanized with APTES and measured with AFM. The black spots are holes which are apparently due the acid cleaning. The deepness of the holes was not possible to determine since the AFM cantilever did not fit in to the holes. b)With APTES silanized surface treated with sulfo-SIAB.

The periodic structure that is punctually visible gives a hint that the sulfo-SIAB forms a monolayer at the glass surface. The root mean squared roughness was determined to be ∼ 0.5 nm in both of the samples and this result verifies that the coverslips have a very good surface quality. The field of view: 1µm×1 µm.

6.5 Conclusions

In this chapter we wanted to show that measurable light scattering signal can be detected from the end-grafted DNA carpets. Therefore we have build an evanescent wave dynamic light scattering (EWDLS) setup which we characterized in terms of penetration depth and diffusion of colloids close to a wall. Additionally we used a data analysis method in which additional coupling of local oscillator to the detected scattering signal is not needed to ensure heterodyning. Furthermore the non-linearities of the PMT are also considered in the data analysis. As a result we found that we have measured a dynamic light scattering signal from an end-grafted DNA carpet. The measured signal was proven to originate from DNA as by using endonuclease DNase I which cleaves DNA, the detected signal vanished. Further analysis of the measured data, however, was not possible as it was rather noisy apparently due lack of thermal stabilization of the setup

The goal of the present work was to establish robust and reliable methods for the preparation and characterization of dense carpets of long-chain DNA attached with both ends simultaneously to the surfaces of a surface force apparatus. This provides the perspective of an experimental platform of studying strongly ex-tended DNA with structure-sensitive methods such as X-ray scattering or optical birefringence. This approach should allow to elucidate the (so far unresolved) structural origin of the plateau in the force-extension curve observed in single-molecule experiments, and to study the role of protein binding in large-scale structural changes in genomic DNA.

In comparison to traditional single-molecule force experiments, the controlled stretching of a large ensemble of DNA molecules using a surface-force apparatus imposes stringent conditions to sample preparation: (i) the end-grafting of DNA to the substrate has to be strong enough allowing overstretching of the molecules, (ii) the end-grafting has to be specific for given substrate and finally (iii) in order to obtain measurable X-ray scattering or birefringence signals, the carpets, in addition to the above mentioned criteria, should ideally be prepared at densities at which the DNA forms brushes.

In this work we have thus developed, on the one hand, an experimental route to the preparation of dense, strongly anchored DNA carpets suitable for surface force experiments, and, on the other hand, new methods to characterize tethering density and mechanical stability under external force.

The mechanical stability of DNA-surface links has been improved by us-ing long-chain DNA end-labelled with multiple biotins on one end and a thiol group on the other end. This characterization was done by using direct obser-vation of rupture events under external force in a confocal fluorescence micro-scope coupled to a simple extension device. We succeeded in characterizing the mechanical stability of large ensembles of double-end-tethered DNA, clearly dis-tinguishing single biotin from multiple biotin tethering. In addition we found that the end-modification of long-chain DNA with short oligonucleotides does not always provide the needed mechanical stability for stretching and that strep-tavidin (the biotin-binding protein) is not always strongly enough coupled to the substrate. This lead us to develop PCR (Polymerase Chain Reaction) syn-thesis of long-chain DNA end-labelled with multiple biotins on one end and a

115

thiol group on the other end. The use of PCR, in principle, eliminates the prob-lem of oligonucleotide ligation as the molecule is synthesized completely with its end-modifications. Additionally we have developed alternative end-grafting pos-sibilities for long-chain DNA molecules so circumventing the problems found in streptavidin-surface grafting.

Furthermore we have developed a new reversible combing method which in-dicates enhanced DNA tethering density. This method is based on unspecific electrostatic absorbtion of end-grafted DNA molecules to the surface which is previously elongated by a hydrodynamic flow. Through multiple combing of new DNA fractions, a systematic enhancement of the tethering density can be achieved, possibly allowing to produce DNA carpets at brush densities since the molecules can be released from the surface. Finally, evanescent wave quasi-elastic light scattering has been established as a sensitive tool for the detection of sub-monolayers of DNA tethered to solid substrates. This method might thus provide a new, marker-free method for the quantification of DNA tethering densities.

Das Ziel der vorliegenden Arbeit bestand in der Entwicklung einer zuverl¨assigen Methode der Herstellung und Charakterisierung dichter Teppiche mit langket-tiger DNA, die beidseitig zwischen zwei Oberfl¨achen einer Kraftapparatur end-angeheftet ist. Damit wurde eine experimentelle Plattform geschaffen, um

¨uberstreckte DNA mit strukturaufkl¨arenden Methoden wie R¨ontgenstreuung oder optischer Doppelbrechung untersuchen zu k¨onnen. Mit dieser Methode soll der bisher unbekannte strukturelle Ursprung des Plateaus in den Kraft-Ausdehnungs Messungen an einzelnen Molek¨ulen aufgekl¨art werden. Weiterhin ist der Einfluss der Proteinbindung bei langreichweitig Struktur¨anderung in DNA von Interesse.

Im Vergleich zu traditionellen Einzelmolek¨ul-Kraftexperimenten, ben¨otigt das kontrollierte Strecken eines grossen Ensembles von DNA Molek¨ulen mit-tels Oberfl¨achen-Kraft Apparatur, strenge Bedingungen der Proben Herstellung:

(i) die Endanheftung der DNA zum Substrat muss stabil genug sein um das Uberstrecken der Molek¨ule zu erm¨oglichen. (ii) Die Bindung muss spezifisch am¨ Substrat erfolgen. (iii) Um R¨ontgenstreuung oder Doppelbrechung des Teppiches mit obengenanten Kriterien zu erzielen, sollten die Molek¨ule idealerweise dicht gepackt auf dem Substrat angeheftet sein (brush regime).

In dieser Arbeit haben wir eine experimentelle Methode entwickelt, welche alle diese Kriterien erf¨ullt. Weiterhin wurden neue Methoden zur Messung von Molek¨uldichten und der mechanischen Stabilit¨at von Bindungen unter Kraftan-wendung entwickelt.

Die mechanische Stabilit¨at der DNA-Oberfl¨achen Bindung konnte durch den Einsatz von multi Biotin an einem und Thiol am anderen Ende modifizierter langkettiger DNA verbessert werden. Die Charakterisierung erfolgte durch di-rekte Beobachtung der Abreißereignisse unter Kraftanwendung im Konfokalen Fluoreszenz Mikroskop, mit Hilfe eines einfachen Streckaufbaus. Dadurch gelang die Charakterisierung der mechanischen Stabilit¨at eines großen Ensembles dop-pelseitig angehefteter DNA mit klaren Unterschieden zwischen jeweils einzelner und multipler Biotinmodifikation an deren Ende. Außerdem konnten wir fest-stellten, dass die Endmodifikation langkettiger DNA mit kurzen oligonukleoti-den nicht immer die ben¨otigte mechanische Belastbarkeit f¨ur Streckexperimente gew¨ahrleistet, und dass Streptavidin (das biotin bindende Protein) nicht immer

117

fest genug an das Substrat gekoppelt ist. Diese Erkenntnisse f¨uhrten zu der Poly-merase Ketten Reaction (PCR) Synthese langkettiger DNA endgelabelt mit mul-tiplen Biotinen an einen Ende und einer Thiolgruppe am anderen Ende. Durch die Einsatz der PCR wurden die Probleme der Oligonukleotid Ligation beseitigt, da wir die Molek¨ule komplett mit ihren Endmodifikationen synthetisierten. Außer-dem umgingen wir die Probleme der Streptavidin-Oberfl¨achen Bindung durch die Entwicklung alternativer Endanheftungsmethoden.

Es wurde eine neue reversible Combing Methode entwickelt, mit deren Hilfe die Endanheftungsdichte von DNA erh¨oht werden kann. Diese Methode basiert auf dem Prinzip der unspezifischen elektrostatischen Absorption an der Oberfl¨ache von endangehefteter DNA, die zuvor durch hydrodynamischen Fluss gestreckt wurde. Durch das wiederholte Combing neuer DNA Fraktionen, kann die Endanheftungsdichte systematisch erh¨oht werden. Eventuell erm¨oglicht dies die Herstellung won DNA Teppichen im ”brush” Regime, da sich die Molek¨ule von der Oberfl¨ache wieder aufstellen lassen. Abschließend ist mit der Evaneszenten quasi-elastischen Lichtstreuung eine empfindliche Methode zur Untersuchung von sub-monolagen angehefteter DNA auf festen Substraten Entwickelt worden. Mit dieser Methode kann eine neue quantitative Dichtebestimmung von angehefteter DNA ohne Marker erreicht werden.

List of chemicals and material suppliers

Antibleaching:

• 1,4-Dithiothreit (DTT): Roche Diagnostics (Cat. No. 708984)

• Catalase from bovine liver: Sigma (Cat. No. C40)

• Glucose Oxidase from Aspergillus niger Type VII:Sigma (Cat. No.

G2133)

• D(+)-Glucose: Merck Antibodies and Enzymes:

• Bovine Serum Albumin (BSA): New England BioLabs (Cat. No.

B9001S)

• DNase I: New England BioLabs (Cat. No. M0303S)

• Streptavidin: Roche Diagnostics (Cat. No. 1721666)

• T4 DNA Ligase: MBI Fermentas (Cat. No. EL0016)

• T4 DNA Ligase: New England BioLabs (Cat. No. M0202S)

• T4 DNA Ligase: Roche Diagnostics (Cat. No. 481220) Buffers and buffer components:

• di-Potassium hydrogen phosphate: Riedel-de Ha¨en (Cat. No. 04248)

• Potassium phosphate monobasic: Riedel-de Ha¨en (Cat. No. 04243) 119

• Potassium chloride: Merck

• Natrium chloride: Carl Roth (Cat. No. 3957.1)

• TRIS borate-EDTA Buffer Concentrate, long run: Fluka (Cat. No.

93290) Cross - linkers:

• Succinimidyl-4-(N-maleimidomethyl)cyclohexane-1-carboxylate(SMCC):

Pierce Biotechnology (Cat. No.1721666)

• N-Succinimidyl(4-iodoacetyl)aminobenzoate(SIAB): Pierce Biotech-nology (Cat. No.22327)

• EZ-Link^r (Pierce, No.:21335): Pierce Biotechnology (Cat. No.22327)

• 1,1´-Carbonyldiimidazole (CDI): Sigma-Aldrich (Cat. No.115533)

• 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide (EDC): Fluka (Cat.

No.03449)

• Glutaraldehyde: Polysciences (Cat. No.00216) Dot-blot:

• Nylon Membranes, positively charged: Roche Diagnostics (Cat.

No.1209272)

• Streptavidin HRP Conjugate (ELISA grade): Biosource International (Cat. No. SNN2004)

Dyes and fluorescent antibodies:

• Cy3-modified oligomers: Thermo Electron

• Streptavidin, Alexa Fluor^r 488 conjugate: Molecular Probes (Cat.

No. S-11223)

• Streptavidin, Alexa Fluor^r 555 conjugate: Molecular Probes (Cat.

No. S-21381)

• YOYO-1: Molecular Probes (Cat. No. Y3601) Gel electrophoresis:

• Agarose - Separation > 500 bp: USB (Cat. No. R75817)

• Ethidium bromide: Amersham Bioscience (Cat. No. 17−1328−01)

• 6X Loading Dye Solution: MBI Fermentas (Cat. No. R0611) Glass cleaning and silanization:

• 3-Amino-propyltriethoxysilane (APTES): Fluka (Cat. No. 09324)

• (3-Glycidyloxypropyl)trimethoxysilane (GOPS): Fluka (Cat. No.

50040)

• Acetic acid: Carl Roth (Cat. No. 3738.1)

• Ethanol: Riedel-de Ha¨en (Cat. No. 32205)

• Hydrogen peroxide 30 %: Merck (Cat. No. 107298)

• Methanol: Riedel-de Ha¨en (Cat. No. 32213)

• Succinic anhydride: Fluka (Cat. No.14089)

• Sulfuric acid 95 % -- 97 %: Riedel-de Ha¨en (Cat. No. 30743) Glassware:

• Microscope cover slips: 24×60 mm, #1.5 Gerhard Menzel, Glasbear-beitungswerk

• Microscope slides: 76×26×1 mm, Marienfeld (Cat. No. 1000000) Nucleic acids:

• λ-DNA: MBI Fermentas (Cat. No. SD0011)

• λ-DNA: New England Biolabs (Cat. No. N3011L)

• Oligomers (single modification): Thermo Electron

• Oligomers (multi modifications): IBA GmbH

• λ-DNA/EcoRI+HindIII Marker 3: MBI Fermentas (Cat. No.

SM0191/2/3) Miscellaneous:

• Adenosine 5´-triphosphate disodium salt (ATP): Sigma (Cat. No.

A2383)

• DMSO: Merck (Cat. No. 107298)

• Gold 99.95%: Goodfellow (Cat. No. AU005171/65)

• NICK^{T M} columns: Amersham Biosciences (Cat. No. 17−0855−01)

• PEG 3350: Sigma-Aldrich (Cat. No. P3640)

• PEG 8000: USB (Cat. No. 19959)

• Poly(acrylic acid): Aldrich (Cat. No. 323667)

• TWEEN^r 20: Fluka (Cat. No.93773)

• Two component - silicone rubber: Carl Roth (Cat. No. 5985.1)

• Spermidine tetrahydrochloride: Sigma (Cat. No. S2876) PCR:

• dNTP Mix (Deoxynucleotide Mix): Eppendorf (Cat. No. 0032003.001)

• Long PCR Control set (for 20 kbp DNA): MBI Fermentas (Cat. No.

K0201)

• Long PCR enzyme mix: MBI Fermentas (Cat. No. SD0011)

• MgCl₂: MBI Fermentas (Cat. No. K0181)

• MinElute^{T M} Gel Extraction Kit (50): Qiagen (Cat. No. 28604)

• End-modified primers: Thermo Electron

• Taq DNA Polymerase: Eppendorf (Cat. No. 0032002.200)

• Template Bluescript 2KS: a gift from Dr. T. Meergans, University of Konstanz

Streptavidin thiolation:

• Hydroxylamine•HCl: Pierce Biotechnology (Cat. No. 26103)

• SATA: Pierce Biotechnology (Cat. No. 26102)

• Streptavidin: Roche Diagnostics (Cat. No. 1721666)

• Zeba^{T M} Desalt Spin Columns: Pierce Biotechnology (Cat. No. 89882)

Functionalizing λ - DNA with oligos

Ligation of oligos:

• Pipette 66 µl ofλ-DNA stock (NEB) in a cup.

• 15 minutes in heat block at 75^◦C.

• 5 minutes cooled in ice.

• Set heat block to 50^◦C.

• Add 10 µl of Ligase buffer (10×).

• Add 3.4µl of thiol, amino or biotin(5⁰ - GGG CGG CGA CCT - 3⁰).

• Add 3.4µl of thiol, amino or biotin(5⁰ - AGG TCG CCG CCC - 3⁰).

• Add 17.2 µl ofMilli-Q water.

• Incubate for one hour at 50^◦C.

• Add 3.4µl of T4 Ligase.

• Add 1 µl of ATP (end concentration 10 mM).

• 30 minutes at 25^◦C or one hour at room temperature.

Precipitation of DNA with PEG:

• Add PEG-8000 and NaCl to the DNA stock in end concentration of 10%

(w/w) (PEG) and 0.5 M (NaCl).

• Mix and put in ice for 30 minutes.

123

• Centrifuge 15 minutes with maximum speed (15000×g) at room tempera-ture (Eppendorf centrifuge).

• Remove supernatant.

• Add carefully ethanol (70%) and remove supernatant.

• Add carefully ethanol (96%) and remove supernatant.

• Let pellet dry in SpeedVac^r or at room temperature.

• Dissolve again with buffer (TBE).

Filtering with NICK^{T M} column:

• Pour off the solution from the column.

• Fill once with TBE and pour off.

• Let 1 ml of TBE flow through.

• Add DNA to the column.

• Add 400 µl of TBE and let it sink in to the column.

• Add 400 µl of TBE and collect the sample under the column.

Also other buffers than TBE can be used.

B.1 FIGE protocol

• Fill the gel tray so that the buffer (0.5× TBE) is 1 mm over the gel.

• Use thermostat to cool the buffer to 11^◦C.

• Adjust the power supply so that the potential difference between the elec-trodes is roughly 160 V. The current should be then roughly 70−80 mA.

• Let the DNA run into the gel for 10 minutes with constant current.

• Run 6 hours with a program H1: forward/backward pulse ratio 3 : 1 with increasing pulse duration from 20 to 30 seconds over the program run time.

• Run 6 hours with a program H2: forward/backward pulse ratio 3 : 1 with increasing pulse duration from 10 to 20 seconds over the program run time.

• Run 6 hours with a program H2: forward/backward pulse ratio 3 : 1 with increasing pulse duration from 0.8 to 10 seconds over the program run time.

• Incubate some 15 minutes in mixture of ethidium bromide and water (con-centration 0.1−0.5 µg/ml).

• Wash few times with water.

• Film under UV light illumination.

B.2 Functionalization of glass slides

Cleaning of glass slides:

• Rinse slides with isopropanol (tech. grade), aceton (tech. grade) and milli-Q water respectively.

• Sonicate for 30 minutes (slides in milli-Q water).

• Prepare 150 ml of 1 : 1 (v/v) mixture of sulfuric acid (95 % – 97 %) and hydrogen peroxide 30% (Piranha water).

• Soak the slides in Piranha water for 30−60 minutes.

• Wash thoroughly first with milli-Q water and then with ethanol (p. a.).

• Store in ethanol (p. a.).

Amino terminated slides (1):

• Mix 2.5% of APTES and ethanol (p. a.) (v/v) (150 ml end volume).

• Transfer the slides to APTES silanization solution.

• Let react for two hours at room temperature.

• Wash with ethanol (p. a.) and milli-Q water respectively.

• Dry in the oven at 75^◦C.

Amino terminated slides (2):

• Prepare a mixture of 1% of APTES, 94% methanol (p. a.) and 5% of milli-Q water.

• Add 10 µl of acetic acid (end concentration of 1 mM in 150 ml) to the silanization solution.

• Transfer the slides to APTES silanization solution.

• Let react for two hours at room temperature.

• Wash with ethanol (p. a.) and milli-Q water.

• Dry in the oven at 75^◦C.

Epoxy terminated slides:

• Prepare a mixture of 2.5% of GOPS and 97.5% ethanol (p. a.).

• Add 10 µl of acetic acid (end concentration of 1 mM in 150 ml) to the silanization solution.

• Transfer the slides to GOPS silanization solution.

• Let react 1−24 hours at room temperature.

• Wash with ethanol (p. a.) and milli-Q water

• Dry in oven at 50^◦C Carboxyl terminated slides:

• Dissolved 0.9 g of succinic anhydride in 666 µl of dimethylformamide.

• Dissolve 2.1 ml of APTES with 1.2 ml of ethanol (p. a.).

• Mix the two solutions together.

• Let react overnight at room temperature.

• Fill the mixture to 150 ml.

• Transfer the slides to the silanization solution (the slides arestaying in ethanol after acid cleaning).

• Let react for two hours at room temperature.

• Wash with ethanol and milli-Q water.

• Dry in oven at 75^◦C.

B.3 Thiolation of streptavidin

• Dissolve SATA in DMSO at a concentration of 108 mg/ml.

• Dissolve streptavidin (lyophilizated powder) 1 mg in 1.0 ml of milli-Q water.

Store frozen at −20^◦C.

• Take 150 µl of streptavidin solution (concentration 1 mg/ml) and change the buffer to PBS with 10 mM EDTA at pH 7.2−7.5 by using Zeba^{T M} spin columns.

• Add 1.5µl of the SATA solution (freshly prepared) to 150µl of streptavidin solution (now in PBS) and let react for 30 minutes at room temperature.

• Wash the reactants away and change the buffer to PBS with 10 mM EDTA at pH 7.2−7.5 by using Zeba^{T M} spin columns. (The product solution can be stored frozen at −20^◦C.)

• Prepare a buffer with 0.5 M hydroxylamine and 25mM EDTA in PBS at pH 7.2−7.5.

• Add the prepared buffer to the streptavidin solution in ratio 1 : 10, respec-tively.

• Let to react for two hours at room temperature.

• Wash the reactants away and change the buffer to PBS with 10 mM EDTA at pH 7.2−7.5 by using Zeba^{T M} spin columns. (The product solution can be stored frozen at −20^◦C.)

Partial heterodyning formulas in auto- and crosscorrelation mode

C.1 Autocorrelation mode

hn₀n_τi

hni² = 1 + X4

i=1

fi(², φ, jl, jb)[g⁽¹⁾]ⁱ (C.1) where

f₁ = 2(1−j_b−j_l)j_l(1−2²[1 + (1−j_l)²−j_b²] +²²{−1 + 4φ+ 2(1−j_b−j_l)(−1 +j_b−j_l +6φ) + (1−j_b−j_l)²[(1 +j_b−j_l)(−1 + 3j_b

−3j_l) + 4(2 +j_b−2j_l)φ]}), (C.2)

f₂ = (1−j_b−j_l)²{1−2²[2 +j_l²−j_b²] +²²[−1−2

×(1−j_b)²+ 10j_l²+ (1−j_b+j_l)(1−j_b−j_l)

×(−1 + 2j_b+ 3j_b² −3j_l²) + 4φ(6−6j_b

+j_b³−3jbj_l²−2j_l³]}), (C.3)

f₃ = 16²²(1−j_b−j_l)³j_l, (C.4)

f₄ = 4²²(1−j_b−j_l)⁴ (C.5) 129

and

² = rθ+r²θ²[1 + (1−j_b)²−j_l²]. (C.6)

C.2 Crosscorrelation mode

hn10n2τi

hn₁ihn₂i = 1 + X4

i=1

f_i(²₁, ²₂, φ₁φ₂, j_l, j_b)[g⁽¹⁾(τ)]ⁱ, (C.7) where

²x = ²₁+²₂

2 , ²²₋ =²²₁ −²1²2+²²₂, E₀ = ²₁²₂

²²₋ , E₁ = ²²₁+²²₂

²²₋ , E₂ = ²²₁−²₁²₂+²²₂

²²₋ , p_x = ²²₁φ₁+²²₂φ₂

2²²₋ , (C.8)

f₁ = 2(1−j_b−j_l)j_l(1−2²_x[1 + (1−j_l)²−j_b²] +²²₋{−1 + 4p_x+ 2(1−j_b−j_l)

×[−1 +E₀(j_b−j_l) + 6p_x] + (1−j_b−j_l)²

×[−1 + 2E₀(j_b −j_l) +E₂(j_b−j_l)²

+4(2 +j_b−2j_l)p_x]}), (C.9)

f₂ = (1−j_b−j_l)²{1−2²_x(2 +j_l²−j_b²)

²²₋[−4−4E₀(j_b² −2j_l²) + 2E₁j_b(2−j_b² +j_l²) +E2(j_b²−j_l²)²+ 4px(6−6jb+j_b³

−3jbj_l²−2j_l³)]}, (C.10)

f3 = 16²1²2(1−jb−jl)³jl, (C.11)

f₄ = 4²₁²₂(1−j_b−j_l)⁴, (C.12)

²₁ = r₁θ₁+r₁²θ₁²[1 + (1−j_b)²−j_l²] (C.13) and

²2 = r2θ2+r²₂θ²₂[1 + (1−jb)²−j_l²]. (C.14)

Characterizing the single-mode fiber and the PMT

In order to characterize the single-mode fiber and the nonlinearities of the PMT, we followed the analysis provided by Flammer et al. [77] and did a standard DLS with 60 nm latex colloids at 90^◦ by varying the laser intensity. Each measure-ment was analyzed with a cumulant fit which gives us the intercept of the field autocorrelation function with the given count rate. This is presented in Fig. D.1 (a) for both of the PMTs of our detector. Now by doing a polynomial fit to the measured data the parameters M, θ and φ can be calculated out of equation (6.27). The results for the fiber and the detector are summarized in the Table D.1.

The physical meaning of the parameters φ and θ is presented in Fig. D.1 (b) where on the row (i) is the input sequence of the PMT. On row (ii) is a non-paralyzable photon counting system shown, meaning that during the dead-time collected photo count do not affect the PMT whereas on the row (iii) the photon counting system is paralyzable and so during the dead-time collected photo count adds up to the dead time. The parameter φ is 0 when the the photon counting system is perfectly non-paralyzable and φ = 1 when the system is perfectly paralyzable. The parameter M is the effective number of modes transmitted by

Table D.1: The non-linearity parameters M, θ and φ. M is the effective mode number of the single-mode fiber,θ describes the dead time of the PMT andφ is the updating parameter (see Fig. D.1 (b)).

Channel M θ φ

0 1.021±0.001 (22.1 ±0.8) ns 0.20±0.06 1 1.012±0.002 (26.8 ±1.4) ns 0.50±0.05

133

(a)

(i)

(ii)

(iii)

t

t

t q

f=0

f=1

(b)

Figure D.1: a) By varying the laser intensity while measuring correlation func-tions with 60 nm colloids at 90^◦ we were able to establish a relation between the count rate and the intercept of the autocorrelation function. By fitting a second order polynomial to the measured data and by using the equation (6.27) we can find out the effective mode number M of the single-mode fiber and the nonlinearity parametersθ andφ of the PMT (see theory and Table D.1). b) The physical meaning of the parametersθandφ: (i) The input sequence coming to the PMT. (ii) θ is the dead-time of the PMT. For non-paralyzable detector (φ = 0) the photon coming during the dead time has no influence to the dead-time. (iii) Photons coming during the dead time lengthen the dead time and so eventually can paralyze the detector: paralyzable detector (φ = 1) (Figure adapted from Sch¨atzel et al. [82]).

the fiber. Our measurements were done with a polarization maintaining fiber¹ and later we changed to another single-mode fiber² but the measurements were

the fiber. Our measurements were done with a polarization maintaining fiber¹ and later we changed to another single-mode fiber² but the measurements were

Im Dokument Mechanical Streching and Lightscattering on DNA (Seite 110-145)