A dual-trap optical tweezer setup for single molecule manipulation : development and testing

Advisors

Prof. Felix Ritort

Departament de F´ısica Fonamental Facultat de F´ısca

Universitat de Barcelona

Prof. Antonio DiCarlo

LaMs Dipartimento di Strutture Universit`a Roma Tre

i ⊗ j denotes the double tensor ¯J such that ¯Jk = (i · k)j . . . 16 h·i denotes the equilibrium average of a given observable . . . 50 h·ii denotes a conditional average, usually the condition is specified in the main text . . . 72

1 Introduction 7

1.1 DNA polymers . . . 7

1.1.1 DNA folding. . . 10

1.2 Optical forces and physics at the micron scale . . . 12

1.2.1 How to measure optical forces . . . 14

2 The Double-Trap Optical Tweezers 21 2.1 Advantages of the DTOT . . . 22

2.2 Limitations of the DTOT . . . 24

2.3 Silica–DNA interaction . . . 26

2.4 Calibration of the Minitweezer . . . 32

2.4.1 Calibration of the DTOT . . . 33

2.4.2 Isotropic calibration . . . 36

2.4.3 Anisotropic Calibration . . . 37

2.4.4 Results . . . 38

2.5 The trap shape . . . 41

3 Equilibrium experiments in the two–trap setup 47 3.1 The linear model . . . 47

3.1.1 Dynamic correlations in the linear model . . . 54

3.2 Direct measurements of the hydrodynamic coupling . . . 55

3.2.1 Removal of reflection effects . . . 56

3.3 Testing the linearized Oseen theory . . . 57

3.4 Measurement of the stiffness of ds DNA . . . 60

3.4.1 High Frequency Measurements with DTOT. . . 60

4 Conclusions and prospects 87

4.1 Pulling in a DTOT . . . 87

4.1.1 Relation to Galilean trasformations. . . 89

A Buffer solutions 93 A.1 TE buffer solution (NaCL 1M) pH 7.5, 100 ml . . . 93

A.2 Citric buffer solution 10mM (Na 30 mM) pH 8, 1l . . . 93

A.3 PBS buffer solution, pH 7.5, NaCl 140 mM, 0.5 l . . . 94

B Half Lambda Protocol 95 C Computation of the correlation function in the hopping model.101 D Pulling Experiments in the linear model 105 D.1 Exponential averages of the work. . . 106

D.2 Relation to Galilean Transformations . . . 110

D.3 Estimates on the relevance of the error . . . 111

D.4 Derivation of the equations . . . 112

Optical Tweezer (DTOT) setup devoted to single molecule experiments. To set the stage for the subsequent discussion, the relevant properties of DNA polymers and optical traps will be reviewed at first.

Particular attention will be devoted to the kind of interactions that are tested in force unfolding experiments and to the force measurement technique which is employed.

1.1

DNA polymers

DNA is a polymeric molecule whose biological relevance needs not be ex-plained. In vivo DNA is essentially found in its double stranded form, in which two polymeric chains bind with each other to form a double helix structure. Nevertheless in our present discussion it is meaningful to start describing single stranded DNA (ssDNA), which is the “atom” of our dis-cussion, in the etymological sense: the structure we cannot break. ssDNA is also a polymeric molecule, a hetero–polymer in chemical language, meaning that it is formed by monomers of different species. There are four types of monomers forming ssDNA (as shown in Figure 1.1). Each of these monomers is formed by a hydrophilic sugar, a hydrophobic nitrogenated base and a phosphate group. The chemistry of these molecules (nucleotides) is quite complex. Only some of its most important features will be discussed here, the most relevant being the kind of chemical bond which joins the different monomers: a covalent bond formed between one of the carbons of the sugar ring of a nucleotide and one of the oxygen atoms of the phosphate group of the next nucleotide (see Fig. 1.2). When bound, adjacent sugar groups are

Figure 1.1: Chemical structure of ssDNA monomers: A) Each monomer is com-posed by a sugar molecule (pale red), linked to a phosphate group (on the right of the sugar) and a nitrogenated base. B) The four different nitrogenated bases which are present in DNA.

separated by approximately 0.6 nm [2]. The sugar backbone formed by a long chain of monomers has two important physical characteristics: it is highly flexible and, when immersed in water, it has a huge negative charge, since the phosphate groups at physiological acidity conditions are fully dissociated. Charged molecules are mostly hydrophilic, and indeed the sugar-phosphate backbone is highly hydrophilic [2]. The flexibility of the chain is due to the fact that the covalent P-O bonds in the phosphate groups have no preferred orientation, so that each link along the backbone behaves as a spherical hinge. It is thus reasonable to regard ssDNA as a freely jointed chain.

However, the main message I want to convey is that the monomers form-ing ssDNA are joined by covalent bonds. The energy needed to break a covalent bond ∆G0

c can be 400 kJ/mol, which equals 160kbT per bond at room temperature (with kb the Boltzmann constant) or 40 pN·nm, while the typical bond length ` of the order of the ˚A. Assuming the activation energy ∆G‡(see Figure 1.3) to be of the same order of magnitude of the binding free energy, the rupture force f∗ for a covalent bond may be roughly estimated as: f∗ = ∆G‡ ` ' 40 pN · nm 0.1 nm = 400 pN.

This important estimate implies that in the force range of the DTOT (1-20 pN) covalent bonds do not break off, so that ssDNA is the ”atom” of our

Figure 1.2: Schematic representation of a four-nucleotides ssDNA filament. Differ-ent monomers are connected through a phosphodiester bond linking the 3’ carbon atom of one monomer to the 5’ carbon atom of the next monomer through the phosphate group. All monomers in the filament have the same orientation, thus endowing the ssDNA a filament with a distinguished polarity.

rapidly with the distance between the atoms involved (as 1/r for dipolar interactions and as 1/r6 _{for van der Waals). The low binding energy of} covalent interactions explains the different stability of covalent and non-covalent structures. Another essential point involves the activation energies needed to form such bonds. Activation energies can be understood in the framework of Transition State Theory (TST), which postulates the existence of a Transition State (TS) along the reaction coordinate, whose energy is higher than that of the reactants and products (see Figure 1.3). In this framework, the activation energy is the energy the reactants participating in a chemical reaction have to acquire to overcome the transition energy gap ∆G‡, while the binding energy ∆G0 _{is the free-energy difference between} re-actants and the product. The activation energy, acquired in a first stage and then released in the course of the reaction, does not enter in the final ther-modynamic balance. However the reaction rate does depend exactly on this energy. Covalent bonds have a high activation energy [1], so that they have a very low reaction rate at room temperature: their formation requires help from catalysts or enzymes. This is precisely the case of the phosphodiester bond formation, which is performed by a specific enzyme, DNA ligase (see Appendix B). Non-covalent bonds, having a low activation energy [1], can form and break off spontaneously at room temperature. As a consequence, non-covalent structures can acquire a reasonable thermodynamic stability only by forming multiple non-covalent interactions. Moreover, due to the short ranged nature of non-covalent interactions, two regions of one molecule can form a stable non-covalent interaction only if their surfaces present some complementary areas in which many atoms of one region can be put in close contact with atoms from the other region [1]. This explains the selectivity of base pairing in DNA and the molecular recognition between ligand and substrates in enzymes.

1.1.1

DNA folding.

DNA is well known to form a secondary structure, in which complementary strands arrange themselves in a double helix. This structure is energetically

Figure 1.3: Free energy profile along the reaction coordinate in the Transition State Theory for chemical reactions. The two relevant energies are the free energy difference between the reactants and products ∆G0 _{and the free energy barrier}

∆G‡ which must be overcome to perform the reaction. While the relative ther-modynamic stability of products and reactants depend only on ∆G0, once this is fixed the kinetic stability is determined by ∆G‡

favored by hydrogen bonding between bases and hydrophobic interactions, both these effects are of non-covalent nature. Hydrophobic effects are due to the free energy gain (or entropy cost) obtained preventing contact between water molecules and hydrophobic bases. Think of two complementary ssDNA strands, with hydrogen bonds formed between the bases of the two strands and arranged in a ladder structure. The space between adjacent bases in the ladder (' 0.6 nm) is large enough to allow water molecules to slip in between the bases. Imagine now to twist the two backbones, thereby forming a double spiral. As the twist increases the bases are brought closer together and the water molecules are kept out. This is the physical mechanism producing the famous double helix. Le us now think of a short ssDNA molecule, whose sequence is comprised of 16 monomers:

CGCAT CGAAAGAT GCG.

Notice that the 6 first bases are complementary to the last 6. Predictions for the folding of ssDNA can be obtained from the web server Mfold [4]. In the case under examination Mfold predicts, at room temperature and 1 M Na+, the formation of an hairpinned structure, basepaired in the complementary regions, ending in a four nucleotide loop. Surprisingly enough, the predicted

overcome. What is the force needed to activate this process? A simple answer can again be estimated by assuming that the activation energy is of the same order of magnitude of the folding energy. The rupture force f∗ can then be estimated as the ratio between the folding energy and the change in the end-to-end distance between the folded (d) and the unfolded (`) configurations (see Sect. 3.5.6 for details). Taking 0.6 nm for the length of one monomer in the unfolded hairpin and 2 nm for the diameter of the folded hairpin.

f∗ = EF

` − d ' 2 pN,

which is well inside the force range we can explore. In section 3.5 we will show how mechanical manipulation of single DNA hairpins can be used to retrieve thermodynamic and kinetic information about the folding process.

1.2

Optical forces and physics at the micron

scale

In the previous section we saw how in principle it is possible to study the folding process of a single DNA polymer by manipulating it with forces in the pN range. We now focus on how it is possible to do this in practice. Manipulation of single molecule is only possible at the present stage when it is coupled in a specific way to some larger scale objects on which a controlled force may be applied. This is the case for the three main force-spectroscopy techniques: AFM, Optical Tweezers and Magnetic Tweezers. In the case of optical tweezers the two ends of the polymer are chemically linked to two micron-sized dielectric beads. The beads are then trapped, either optically or by air suction on a micron-size pipette, so that the the molecule can be stretched by distancing the two trapped objects. It is to be stressed that the bonds connecting the molecule to the beads should be of non-covalent nature, as they must spontaneously form in a microfluidic chamber.

There is a huge scale separation between the molecule and the beads to which it is linked. Should we scale the molecule and the linker up to the size of a 1 cm thick and 10 cm long rope, the bead radius would become 10 meters.

sphere happens at low Reynolds numbers the equation of motion for such system is:

m ˙v = −γv + f, (1.1)

with v the speed of the sphere, m = 4₃πa3ρm and γ = 6πηla. The Reynolds number Re, defined as:

Re = aρlv ηl

,

is the ratio of inertial to viscous forces. There is an easy way [3] to estimate the value of Re in the case we are considering: if we take the viscosity of the liquid, we square it and divide it by the density of the liquid then we obtain a force. This is the force which will push anything, independently of its size, at Reynolds numbers close to one. In the case of water,

η2 l ρl

= 10−9N,

implying that the flow around the ball will happen at low Reynolds number if we push it with a force under the nN. Let us now suppose that f is in the order of the pN, so we are sure that the Reynolds number is low and that we can consistently use equation (1.1). The solution of the this equation reads:

v(t) = e−(γ/m)tv0+ (1 − e−(γ/m)t) f

γ, (1.2)

which implies that the ball speed of the sphere will exponentially approach the limit velocity set by the force f . The inverse time scale of this exponential approach equals the ratio of the friction coefficient to the mass:

γ

m =

9ηl 4ρma2

, (1.3)

which means that the timescale on which the sphere reaches the limit velocity goes to 0 as the radius of the sphere is decreased. In the experimental situation we consider, this timescale is τI ' 10−7s, way too fast for data

increases as the length scale decreases. This can be easily seen evaluating the average of the square of a particle’s velocity through equipartition:

phv2_{i =} r

kbT

3m, (1.5)

which shows how velocity fluctuations of the sphere due to thermal noise grow as is mass decreases. This prediction from equipartition theorem is also out of reach of the setup we are going to describe, mainly because the correlation time for the bead’s speed, which is the same τI discussed above, is too small. Nevertheless, equipartition holds for any degree of freedom, so that if we imagine the bead being trapped in a harmonic potential of stiffness k = 1pN/nm, with U (x) = 1₂kx2, we have:

phx2_{i =} r

kbT

k ' 2 nm. (1.6)

Thermal fluctuations define the resolution obtained when manipulating single molecules, so that a way to read (1.6) is: a bead trapped in a harmonic potential of stiffness k = 1 pN/nm can be used to probe phenomena which involve length scale changes on the order of some nanometers. The size of the bead itself does not matter here, only the size of its spatial fluctuations being involved.

1.2.1

How to measure optical forces

We now know that a micrometric object trapped in a harmonic potential can provide the resolution to probe conformational changes in nucleic acids. However the issue of how to trap micrometric objects remains open and the use of light beams is one of the available solutions. This trapping mechanism was discovered and explored by A. Ashkin in the 70’s and has now become a standard technique. I will just recall that a focused laser beam which shines on a micron sized dielectric bead can trap it in an optical potential due to the balance between scattering and gradient forces. What is to be stressed here is that the very same light which is used to form the optical trap will

Figure 1.4: Top: Dashed lines indicate the outermost light rays collected by the objective and the detector. The inner cone shown is the actual beam that enters from the left objective, traps the bead and is imaged on the detector through the right objective. Bottom: the force applied to the bead manifests through a deflection of the light beam traversing the bead. Since the beam underfills the left objective, all the transmitted light is collected by the right objective; while reflected light is not correctly measured. The need to use underfilling beams limits the attainable maximum trapping force in the direction of the optical axis. The deflection of the light can be measured and used to reconstruct the force acting on the bead. Picture was reproduced from [18].

carry the fingerprint of its interaction with the trapped object away from the interaction site and hopefully directly into our detectors. A set up in which this can be done, developed by Smith, Cui and Bustamante [18], is shown in Figure 1.4. Here a light beam enters a microfluidics chamber from the left through an objective which focuses it into a spot, where a bead gets trapped. After the interaction with the bead the light exits from the right through a second objective.

If an external force acts on the particle during its interaction with light, the transmitted light will be deflected. Its deflection can be measured through a Position-Sensitive Detector (PSD) and used, in combination with the law of

the refractive index of the liquid contained in the microfluidics chamber and c is the speed of light. Let us now encircle the bead through an arbitrary spherical surface B of radius R completely contained inside the microfluidics chamber, and let dA be its surface area element. Conservation of linear momentum guarantees that the sum of the total momentum change inside this surface, due to changes in the bead’s momentum and changes in the field’s momentum, must equal the momentum flux through the surface:

d

dt(Pbead+ Pfield) = Fbead+ Ffield = Z

∂B

¯tˆndA, (1.8)

where ˆn is the outward pointing normal to the surface B. There is a large scale separation between the relaxation times of the electromagnetic field and those involved in the motions of the bead. We can thus approximate the electromagnetic field as stationary at any instant on the timescale of the motion of the bead. In this approximation the momentum of the field in the volume B does not change in time so that

d dtPfield = Ffield = 0, and Fbead = Z ∂B ¯tˆndA. (1.9)

The momentum flux can be computed, in the geometric optics approxima-tion, as in (1.7): Fbead = n c Z ∂B ˆ_{d ⊗ (Sin− Sout)} ˆ ndA. (1.10)

Here we decomposed the Poyinting vector in two components, one due to the light entering through the surface B from the objective and the other one to the light exiting from the surface B. We now chose the radius R such that it is much bigger than both the trapped bead’s radius and the distance between the bead’s center and the focus of the incoming radiation. In such case both the Sin and Sout components can be thought as perpendicular to

Fbead =

c _∂B(Sin− Sout) dA

Moreover, since S is normal to the surface we can define an angular intensity, related to S by: I(θ, φ)ˆrdγ 4π = SdA, (1.11) (dγ_4π = dA_R2), so that (1.9) becomes: F = Z dγ

4πI(θ, φ)ˆisin(θ) cos(φ) +ˆjsin(θ) sin(φ) + ˆk cos(θ) , (1.12) with the coordinate θ and φ as defined in Figure 1.5. In order to correctly measure the intensity of the light transmitted through the bead all the light exiting from the interaction zone should be collected. A source of systematic error in this setup is due to reflected light, which does not exit the interaction zone from the same site as the transmitted light. This effect will be discussed in section 2.5.

As far as the transmitted light is concerned, its angular intensity profile can be imaged on the image side principal plane of the secondary objective. This can be done through Abbe’s sine condition, which states that a light ray exiting from the focus of a coma free objective lens, at an angle θ with respect to the optical axis of the objective, but still hitting the lens, will exit image side principal plane of the lens at a distance r from the optical axis, with r given by:

r = RLn sin(θ), (1.13)

where n is the refraction index of the fluid in the object side of the lens and RL the focal length of the lens. By energy conservation the angular intensity exiting through the solid angle sin(θ)dθdφ is projected without loss on the area dA0 on the image side of the lens, and is given by:

I(θ, φ)dγ

4π = E(r, φ)dA 0

Figure 1.5: Schematic representation of the two coordinate system used in the description of the force measurement method.

where E(r, φ) is the irradiance (Watts/m2_{) at the image side principal plane} of the objective lens. Due to (1.14) the force acting on the bead can be reconstructed as: F = 1 c Z dA0E(r, φ)  ˆi r RL sin(φ) + ˆj r RL cos(φ) + ˆk r 1 − r RL  . (1.15) Although it is not possible to predict E(r, φ) it is possible to measure it through Position Sensitive Devices. Such device, placed at the principal plane of the objective lens, returns two signals which are proportional to the integral of the irradiance weighted through the distance to the detector’s center: Dx = ψ Z E(x, y) (x/RD) dA0 = ψ Z E(r, φ) (r cos(φ)/RD) dA0 (1.16) Dy = ψ Z E(x, y) (y/RD) dA0 = ψ Z E(r, φ) (r sin(φ)/RD) dA0, (1.17) where ψ is the detector responsivity and RD its half width. The force act-ing in the plane perpendicular to the optical axis is then know apart from fundamental (c) or measurable (ψ,RL,RD) constants:

Fx= DxRD cψRL (1.18) Fy = DyRD cψRL . (1.19)

been obtained using Double Trap Optical Tweezers (DTOT). The great suc-cess of this setup is due, among others, to two peculiar features. On the one hand the possibility of manipulating single molecules by all optical means guarantees an exceptional isolation from ambient noise, allowing unprece-dented resolutions. In 2005 this led to the measurement of single base pair stepping of RNA polymerases [9] and paved the way for polymerase based sequencing [10]. On the other hand, in many experimental situations, the use of cross correlations between the signals coming from the two traps allows to overcome the resolution limit due to the stiffness of the traps [11],[6], and to measure cross correlations between the two beads. Many crucial experiments have been performed in the last ten years with this kind of setup, e.g. the direct measurement of hydrodynamic correlations between trapped particles [5] and the stiffness of long double stranded DNA molecules [6]. More re-cently the growing resolution and stability of the instruments allowed also the measurement of the full, sequence dependent, free energy landscape of DNA hairpins [7],[12]. In this section a novel set up is presented, which works with counter propagating beams and by direct measurement of light momentum. This set-up is based on the Minitweezers [27], as described in [18], which is originally designed to operate using two counter propagating beams from dif-ferent sources to form a single trap optical tweezer (STOT). In the original design of the Minitweezers single-molecule force spectroscopy experiments can be performed using the trap to hold one end of a molecule and a micro– pipette to hold the other end. We will now show how it is possible to exploit the Minitweezer to form two different optical traps (DTOT) and to perform single-molecule pulling experiments without using the micro–pipette.

Figure 2.1: Single Trap Setup in the Minitweezers (STOT). Two counter propa-gating beams are used to form a single trap. Single-molecule experiments can then be performed manipulating two micron sized bead, tethered by a polymer (DNA, RNA, Protein). One of the beads is held in the optical trap, while the other is kept fixed on a micro–pipette.

2.1

Advantages of the DTOT

The experimental setup we describe in this thesis (Fig. 2.3 A) uses two counter propagating laser beams of 845 nm wavelength to form two optical traps where dielectric beads are trapped by mutually balanced scattering and gradient forces. The set up is similar to the one described by Smith et al [18]. Here two microscope objectives with numerical aperture 1.20 act as focusers and condensers. One laser beam is focused through its objec-tive while the other objecobjec-tive collects the exiting light, which is redirected to a Position Sensitive Detector (PSD), as shown in Figure 2.2. The laser beams having orthogonal polarizations, their optical paths may be separated by using polarized beam splitters. When the optical trap exerts force on a particle, the incoming light beam is deviated. The deviated exiting light is redirected from the back focal plane of the objective to a PSD that returns a current proportional to the force. Since the force measure is based on light momentum conservation, the calibration is independent of the power of the lasers, the refraction index of the micro spheres, their size and shape, the viscosity and refractive index of the buffer.

PBS l/4 l/4 PBS PSD Bulleye Photodetector Bulleye Photodetector PSD

Relay lens Relay lens

LED

Fluidics chamber

Motorized stage

Figure 2.2: Optic Scheme of the Minitweezer setup. The two different laser beams are drawn in green and yellow. The Position Sensitive Devices (PSD) on the upper part of the scheme measure the position of the traps, while PSD on the lower side of the scheme measure light momentum as described in section 1.2.1.

This DTOT setup has some advantages with respect to the original Minitweez-ers configuration, the main one being the fact that single molecule experi-ments can now be performed manipulating the beads just by optical forces, without the use of the micro–pipette. This guarantees a better insulation from ambient noise. The micro–pipette used in the original configuration undergoes mechanical deformations during the experiment leading to drift in the measurements (Fig.2.3 B, upper left panel). Such deformations are uncontrolled: while the position of the trap is measured with nanometer res-olution, the position of the pipette’s tip is unknown. Under these conditions the pipette–to–trap distance Xp is measured from the variations of the posi-tion of the trap T alone (Fig. 2.3 B, lower left panel), assuming the pipette to be quiet. Xp is nevertheless a crucial quantity in many experiments, e.g., pulling experiments, and it should be known with high accuracy. In the DTOT set up the position of both traps is measured with nanometer reso-lution and the trap–to–trap distance is obtained as a difference between two measured quantities (Fig. 2.3 A). Some experimental drift may still arise from hysteretic effects in the piezos, but this can be easily measured and corrected.

The enhanced stability of the DTOT set up allows long and precise mea-surements (Cf. Fig. 2.3 C, where a 10-minute force trace with no drift is

respect to other dual trap optical tweezers. The two traps are obtained from counter propagating beams coming from different sources, and so they are free from the crosstalk and interference phenomena described in [14]. In our set up we use 150 mW lasers, which are low in power if compared to other instruments. This is particularly important in single-molecule experiments as, together with the use of silica beads, it leads to a lower production rate of Reactive Oxigen Singlets (ROS) at the bead’s surface, thus reducing the oxidative damage of the biological tether [13] (Fig. 2.3 B, upper right panel).

2.2

Limitations of the DTOT

The Minitweezers setup uses undefilling laser beams in high Numerical Aper-ture (NA) objectives [27]. This is due to the fact that, in order to mea-sure forces by conservation of light momentum correctly, the highest possible amount of light must be collected after its interaction with the bead. Raising the NA of the beam could possibly cause some of the transmitted light to fall off the angular acceptance of the objective after the interaction with the bead. The low numerical aperture of the beam is also a limiting factor in the performance of the DTOT, as it limits the attainable gradient force. In the case of polystyrene beads (Kisker Biotech, refraction index 1.5) optical trapping proved almost impossible. This is because polystyrene has too high an index of refraction, which leads to a high scattering force. Trapping was instead possible with 4µm silica beads (Kisker Biotech, refraction index 1.3). This low refraction index lowers both gradient and scattering forces, but en-hances their ratio. In fact, the lower refraction index and the use of a single beam lead to far lower stiffness and maximum trapping forces than in the STOT (see section 2.5). Nevertheless in some cases it is possible to overcome these limitations by using the cross correlation of the signal coming from the two traps (see Section 3.1 or [11]). Besides the limited resolution, the use of silica beads also poses a problem of chemical nature, to be discussed in the next section.

Figure 2.3: A) The experimental setup described in this chapter: two optical traps are created from counterpropagating laser beams. The force exerted by the traps is measured with sub-picoNewton resolution by direct measurement of light momentum, while the position of the trap is monitored with nanometer accuracy. B) Experimental advantages of the DTOT setup. Upper left panel: the DTOT set up is free from drift effects introduced by the micro–pipette. Lower left panel: the measurement of the trap–pipette distance assumes the pipette to be stationary and only considers the displacement of the trap. Upper right panel: the low power of the lasers involved and the use of silica beads reduce the oxidative damage from photo activated Reactive Oxigens Singlets (ROS) [13]. Lower right panel: in the DTOT setup connections leading to shear forces in the molecule are avoided because the bead is free to rotate. C) The great stability of the two-trap setup allows drift free long term measurements, here we show a ten-minute passive mode hopping trace, which is completely free from drift.

acids, making duplex DNA a strong polyelectrolyte carrying a charge of 2e per base pair in most pH conditions. In multivalent polyelectrolyte solu-tions this ultra-high surface charge can be substantially shielded by different cations either by site specific binding or by counterion condensation [16]. Sil-ica surfaces, even if more difficult to characterize, are also negatively charged when in contact with basic solutions. As a consequence, electrostatic inter-molecular interactions between the fixed charges on DNA and silica will then in general disfavor adsorption at low ionic strength. Electrostatic repulsion must then be overcome by a positive mechanism leading to DNA adsorp-tion on silica, which is a conspicuous effect, still poorly understood. Infact silica–DNA interaction is commonly used for biotechnological applications, such as DNA separation [15]. Due to lack of information on the physico-chemical basis of DNA adsorption on silica, no a priori speculation on the optimal buffer solution to be used in single-molecule experiments is available. Adopting an empirical attitude, double stranded DNA pulling experiments where performed in different buffer solutions. Three different buffer solutions at different salt concentrations where tested:

• TE Buffer pH 8.5 NaCl 150mM and NaCl 1M • Citric Buffer pH 8 Nacl 150mM and 1M

• Phosphate Buffer Solution (PBS) pH 7.4 NaCl 150mM and 1M

The protocols to obtain these buffers are reported in Appendix A. A 24kbp dimer form λ phage DNA was tailed at the two ends with biotin and digox-igenin tags (see Appendix B), a solution of such molecules was then incu-bated with 3.5 µm streptavidin coated polystyrene beads (Kisker Biotech) (Fig. 2.4A). The bead/DNA complex was then transferred into a microflu-idics chamber filled with the proper buffer solution and trapped in an optical trap formed by two counterpropagating laser beams. A second bead, this one made of silica and coated with streptavidin (Kisker Biotech) was trapped on a micro–pipette inside the chamber. The two beads were put in contact until a connection was formed (Fig. 2.4B). The trap was then pulled apart at constant velocity while measuring the force signal (Fig 2.4C). The force and

polymer in fully extended configuration, shows no anomaly. In TE buffer and in Citric Buffer DNA adsorbed on Silica independently of the salt con-dition. On the contrary, no adsorption occurred in PBS and smooth curves were obtained both when stretching and when relaxing. Again, also in PBS salt conditions do not matter and smooth curves are obtained at 1M NaCl as well as at 150mM NaCl. The three buffer solutions, which are very similar in pH and ionic strength, differ in the nature of the cations found in the solution. Indeed, the TE buffer is based on tris(hidroximetil)aminomethane, (HOCH2)3CNH2, which has only one dissociable proton with pKa 8.06 [17]. In TE buffer we have thus univalent cations. Citric Buffer and PBS are based on citric acid and phosphoric acid respectively. Both these acids are triprotic [17] ( i.e. can donate three protons); the pKas of the donor groups are listed in Table 2.1. The pH conditions of the buffer I tested are such that while the citric acid was completely dissociated at pH 8, giving triva-lent cations, the phosphoric acid was only partially dissociated (the third pKa is above 12 Tab. 2.1), giving divalent cations. It is thus reasonable to conjecture that it is the valency of the cations present in the solution which determines the strength of silica–DNA interaction. A further proof of the fact that silica–DNA interaction is strongly reduced or completely eliminated in PBS solution is given in Chapter 3.5, where the folding thermodynamics of a DNA hairpin are studied using both polystyrene and silica beads. It must be remarked that DNA adsorption on silica surfaces, while still being an un-desired effect in most experimental situations, is a very interesting and still poorly understood physical phenomenon with implications in physical chem-istry and polymer science [23],[24]. The setup described in Fig. 2.4 could be used to study in detail silica–DNA adsorption at the single-molecule level, in any ambient medium. Data as those shown in Fig. 2.6 could be used for example to obtain a statistics of the length of the rips. We hope to perform such experiments in the near future.

Figure 2.4: The three main steps in a pulling experiment with double stranded DNA. A) Tailed DNA (Appendix B) is incubated in an eppendorf tube with anti-digoxigenin coated polystyrene beads for 15 min. Specific non covalent bonds are formed between the antidigoxigenins on the bead and the digoxigenin on the DNA tail. B) The bead-DNA complex is transferred to a microfluidics chamber where it is trapped by two counter propagating laser beams. The polystyrene bead is then put in close contact with a second, streptavidin coated, silica bead (Kiske Biotech). A second non covalent bond is formed between the biotins on the other DNA tail and the streptavidins on the bead. C) After a contact is formed the trap is pulled apart from the pipette measuring the applied force and the traveled distance. The protocol can be repeated in a cyclic fashion (see Fig. 2.7).

Acid Formula pKa₁ pKa₂ pKa₃

Citric acid C6H8O7 3.14 4.77 6.39

Phosphoric acid HPO4 2.12 7.21 12.32

Table 2.1: Chemical formulas ad pKas for citric and phosphoric acids. The pH of the buffer solutions used in the experiments described in section 2.3 were 8 for the citric buffer and 7.5 for PBS. In these conditions, the citric acid in the citric buffer is completely dissociated, while the phosphoric acid is only partially dissociated (pKa2 < pH < pKa3). See [17] for more details.

Figure 2.5: Force-extension curve of dimer composed by half of the λ phage genome, in T.E. buffer, using polystyrene beads. The corrected WLC force-molecular extension curve from [31] was fitted to the data and is shown in red. Fitting parameters were the persistence length (P) and the Stretch modulus (S)[31],[20] and see Sect. 3.4. The fit values are P = 44±1 nm, S = 1400±200 pN. The fit is carried out on data up to the expected contour length of the molecule. Data show the full overstretching transition, which takes place at ' 65 pN at this salt concentration and it is almost reversible (see the small hysteresis loop at the onset of the transition).

Figure 2.6: The effect of silica–DNA interaction on the stretching curve in pulling experiments on double stranded DNA. Example of stretching/relaxing force-extension curve, the stretching branch (blue) shows sudden drops (black arrows), which can be interpreted as progressive detachment of the DNA molecule from the silica surface. Such rips are also seen at very high forces. At the rightmost limit of the graph it is possible to observe the onset of the overstretching transition, which manifests as a plateau in the force-extension curve. On the contrary, the relaxing branch of the force-extension curve is smooth, and compatible with those obtained with polystyrene beads or in PBS solution. This is compatible with the given in-terpretation of the drops as in the relaxing branch the DNA molecule is initially fully extended and connected to the silica bead only through the specific interac-tion between biotin and streptavidin (Fig.2.4). The sawtooth pattern is observed with both TE buffer and citrate buffers, independently of the salt concentration.

Figure 2.7: Several cycles of the pulling protocol are shown. In the upper panel the measured force is given as a function of time, black arrows marking the points where DNA adsorption manifests. Desorption displays a different pattern of rips in each cycle. The lower panel shows the trap-to-pipette distance on the same timescale. As it can be checked adsorption phenomena only influence the stretching curve.

Fi = MiPSDi+ Fi i = x, y. (2.1) Here Fi is the force exerted by the light on the trapped bead, PSDi is the sum of the readings of both PSD in the i direction in analog to digital units (adu or AU), F0_i is the force offset and Mi is the calibration factor. Calibrating the instrument amounts to a precise measurement of Mi. Since the instrument works by direct measurement of light momentum (see [18] for details), the Mi’s are independent of the experimental conditions, laser power included. The Mis can be obtained by applying a known force on a trapped bead and measuring the PSD response.

In the original setup with only one trap formed by two counterpropagating beams this can be done using stokes law. A micron-sized bead is trapped in a microfluidics chamber filled with water, and the whole microfluidics chamber is moved at a fixed speed v using a motorized stage. The force due to the fluid motion around the bead is known to be

F = 6πηwrv, (2.2)

where ηw is the viscosity of water and r the radius of the bead. The calibra-tion factors can then be obtained combining (2.2) and (2.1). This procedure was performed in the original setup on over 40 beads and the calibration fac-tor was measured with 2% accuracy1 _{(See Supplementary Material in [27]).} This accuracy is granted, among other factors, by the use of 3µm polystyrene calibration beads (Polyscience Inc.) whose radius has very small variance (0.05µm). Using the sum of the readings of the two calibration factors in (2.1) is equivalent to assume the same calibration factor for the two PSDs. This is correct in the STOT but not in the DTOT setup.

The detection of calibration errors in the two-trap system is made easy by the condition that at mechanical equilibrium i.e., when the two traps do not move, the mean force in the two traps must be equal in magnitude and differ-ent in sign (Fig. 2.8). To check this, experimdiffer-ents similar to those described in Section 2.3 were performed measuring force fluctuations at a fixed trap-to-trap distance. The results are reported in Fig. 2.8 where the probability

1_{At this level of accuracy the relation (2.2) is only valid if the trapped bead is hundreds}

The method used to calibrate the two traps separately needs the STOT setup in order to work properly. Single beam trapping is only possible with silica beads whose size and shape are far more disperse than those of polystyrene calibration beads (see sectl 2.2). This makes the usual Stokes calibration inefficient, because the friction coefficient of the beads is known with very low accuracy (within 20%, see Table 2.2). By error propagation this large error on the friction coefficient would affect the precision of the measurement of the calibration factor. However, assuming the STOT setup to be well calibrated, it is possible to measure the friction coefficient γ of each bead through:

F = −γv, (2.3)

where F is the force measured in the STOT and v the speed of the moving stage or, equivalently, the velocity of the fluid with respect to the trap. Repeating the same test with only one of the two traps lit, it is possible to measure the PSD value as a function of v.

PSDA_i = δ_xAvi, i = x, y, (2.4)

where PSDA_i is now the value of the PSD A in the x direction. Combining (2.3) and (2.4) we get:

F_xA = γ δA

x

PSDA_i = ΓA_xPSDA_i (2.5)

The results of the STOT measurement of the friction coefficient on seven different beads is reported in Table 2.2, the force–velocity relation was mea-sured moving the microfluidic chamber along the x and y directions. The values measured along the x and y axis coincide, meaning, to a first approx-imation, that the trapped bead is either spherical or it is free to rotate. The coefficients are obtained by a linear fit to force–velocity curves of the kind shown in Fig. 2.9.

The coefficients δ_ij were measured again during Stokes’ tests, with only one of the two traps lit. The PDS reading was fitted as a linear function of v. The data collected were analyzed in two different ways: in the first case

Figure 2.8: A) When two tethered beads are in mechanical equilibrium, i.e. when the two traps do not move, the mean forces on the two traps must be opposite. Departures from this balance reveal a calibration error. B) Probability distribution of the sum of the signals obtained from the two traps at rest, as obtained using the same calibration factor for both traps. The three curves refer to different values of the average tension in the tether. The error grows with the average tension in the tether as it is to be expected in the presence of calibration errors.

Table 2.2: Friction coefficients as measured in the STOT setup along the x and y axis for 7 different beads. The values are reported in pN· s/µm. Compatible results are obtained by measuring along the x and the y axis. The last column shows the ratio between the bead radius to the radius of the first bead as estimated from Stokes formula. As it is seen errors of the order of 20% are not uncommon.

Figure 2.9: Seven different force–velocity curves as obtained in a Stokes’ test in the STOT. During this test the motor stage moves the microfluidics chamber with respect to the optical trap at constant velocity. During a test the stage moves back and forth along its whole range. The speed grows cycle after cycle.

Table 2.3: Results for the PDS-v relationship obtained from the Stokes’ test. These values are obtained moving the stage back and forth in one direction with only one trap lit (Trap A for the first two columns and Trap B for the second two). The velocity is raised at every cycle.

isotropy in the P SD − v relation was assumed, meaning that the linear fit was performed on the whole velocity interval spanned by the stage during the Stokes’ test. In the second case, the data for positive and negative velocities were fitted separately. In this case, the calibration factor on whether the trap exerts a positive or negative force. This anisotropic behavior of the P SD − v relation can arise if, e.g., the propagation of the laser beam is not perfectly collinear with the measured z axis.

2.4.2

Isotropic calibration

The values of δj_i obtained by isotropic calibrations are reported in table 2.3, while the corresponding calibration factors for each trap separately are re-ported in Table 2.4.

Given the values in 2.4, from the maximum likelihood principle one ob-tains:

µ(ΓA_x) = 1.320 · 10−2(adu/pN ) σ(ΓA_x) = 3 · 10−2(adu/pN ) µ(ΓA_y) = 1.347 · 10−2(adu/pN ) σ(ΓA_y) = 5 · 10−2(adu/pN ) µ(ΓB_x) = 1.488 · 10−2(adu/pN ) σ(ΓB_x) = 5 · 10−2(adu/pN ) µ(ΓB_y) = 1.491 · 10−2(adu/pN ) σ(ΓB_y) = 3 · 10−2(adu/pN ).

These results can be compared with the calibration factors used in the STOT setup, namely:

Table 2.4: The table reports the value for the calibration factors as obtained from seven different beads by isotropic calibration (see text). These calibration factors were obtained combining the values of the friction coefficient as in Table 2.2 and the values of δ_ijas in Table 2.3. As detailed in text the calibration factor is obtained through equation to (2.5).

In particular, one obtains that the average value of the calibration factors obtained independently for each trap are consistent with the value used in the STOT: µ(ΓA x) + µ(ΓBx) 2 = (1.40 ± 0.04)(adu/pN ), µ(ΓA_y) + µ(ΓB_y) 2 = (1.42 ± 0.04)(adu/pN ). In both cases the difference is well below the experimental error.

The corrected force traces obtained with the new calibration factors (F_i0) are obtained as F_i0j = Γ j i Γi F_ij i = x, y j = A, B.

The new distribution of the error on balance is shown in Figure 2.11; the corresponding mean and variance are reported in Table 2.10.

2.4.3

Anisotropic Calibration

In the case of anisotropic calibration, the traces from Stokes’ tests were parti-tioned according to the sign of the velocity of the moving microfluidic cham-ber. Traces for v > 0 and v < 0 were then fitted separately (Fig. 2.10). Anisotropy in the PSD−v relationship, evident in Fig. 2.10, is statistically significant (Table 2.9). The values extracted from the Stokes’ tests are sum-marized in tables 2.5, 2.6,2.7 and 2.7. Each table contains the values of δj_i for both groups (v > 0, v < 0) and the corresponding mean and difference, to

Figure 2.10: Data from a Stokes’ test in the DTOT analyzed assuming different slopes for v > 0 and v < 0. The dark blue curve is the fit for the positive velocity group and the light blue is the fit to the negative velocity group. The anisotropy is statistically significant, as shown in Table 2.9.

help detect when the anisotropy is significant. The resulting calibration pa-rameters are reported in Table 2.9, where it clearly appears that anisotropy is significant in the the x axis for trap A and the y axis for trap B and at the limit of significance in the y axis for trap A. The calibration factors obtained by this method were used to correct the force traces just as in the case of isotropic calibration. The results are discussed in the next section.

2.4.4

Results

In evaluating the accuracy of the calibration factors obtained by direct mea-surement, with the methods described in the previous sections, my guidance will be the basic physical constraint that at rest the average difference be-tween the force signals obtained from Trap A and Trap B should be as close to zero as possible. As clearly seen from Figure 2.11, the Anisotropic calibra-tion, gives the best result. The obtained forcee differences distributions can be fitted to Gaussians whose parameters are reported in Table 2.10, which quantifies the improvement due to anisotropic calibration.

fur-Table 2.5: Coefficients of the PSD−v relationship for Trap A when moving the microfluidic chamber along the x direction. Data from the Stokes’ test (see Fig. 2.10) was analyzed using the anisotropic calibration method discussed in the text. The data from a Stokes’ test were used to derive a linear relation between the speed at which the microfluidic chamber is moved and the PSD response. The data corresponding to positive and negative velocities of the microfluidic chamber were fitted separately. The offset in the PSD−v relation was assumed to be 0. The table reports the slope resulting form the fit to the PSD−v curve obtained in the Stokes’ test for both for positive and negative velocities (see figure 2.10), as well as their difference and mean value. The anisotropy can reach 10% of the average value obtained.

Table 2.6: Coefficients of the PSD−v relationship for Trap A as obtained when moving the microfluidic chamber along the y direction. See the main text or the caption of Table 2.5 for details on the fit procedures. The table reports the slope resulting form the fit to the PSD−v curve obtained in the Stokes’ test for both for positive and negative velocities (see figure 2.10), as well as their difference and mean value. In this case, anisotropy is smaller than Table in 2.5, reaching at most 4% of the average measured value,while the sign of the difference fluctuates. In this case no significant anisotropy is detected.

Table 2.7: Coefficients of the PSD−v relation for Trap B as obtained when moving the microfluidic chamber along the x direction. See the main text or the caption of Table 2.5 for details on the fit procedures. The table reports the slope resulting form the fit to the PSD−v curve obtained in the Stokes’ test for both for positive and negative velocities (see figure 2.10), as well as their difference and mean value. The data show a modest anisotropy, the difference in the measured values is below 4% of the average, nevertheless every the difference of the measured values has the same sign in all the cases.

Table 2.8: Coefficients of the PSD−v relation for Trap B as obtained when moving the microfluidic chamber along the y direction. See the main text or the caption of Table 2.5 for details on the fit procedures. The table reports the slope resulting form the fit to the PSD−v curve obtained in the Stokes’ test for both for positive and negative velocities (see figure 2.10), as well as their difference and mean value. The data show relevant anisotropy, almost reaching 10% of the mean measured value.

seen that the anisotropy is significant in the cases of Trap A in the x direction and Trap B in the y direction. Anisotropy is at the limit of significance in the case of trap A in the y direction.

ther. Indeed one can obtain approximate calibration factors for the DTOT setup from those used in the STOT setup directly from equilibrium experi-ments (e.g. the hopping experiexperi-ments described in 3.5). Let FA and FB be the force signals coming form Trap A and Trap B of an instrument which was calibrated to be used in the STOT configuration, and let cA

i , CiB be the solutions to the following system of equations:

cA_i hF_iAi = cB_i hF_iBi (2.6) cA_i + cB_i

2 = 1. (2.7)

Here the first equation just enforces the physical constraint, while the second equation requires that the average of the calibration factors of the two traps to be equal to the calibration factor for the STOT set up, which is consistent with the results of section 2.4.2. The two equations (2.6) and (2.7) have two unknowns and can be solved. They have an unique solution when the average forces are non zero and of opposite sign. The calibration factors for the independent traps are then obtained as:

ΓA_i = ΓicAi ΓB_i = ΓicBi .

This method, although not based on direct measurements is stems from the same assumptions which were used as guidance during the calibration. It can be used as an expeditious alternative.

2.5

The trap shape

The shape of the optical traps, or, more precisely, the way in which force depends on the distance trap and the bead’s center, can be studied with a

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30

Percentual Error

Probability Density

Single Trap Calibration

Figure 2.11: Comparison of the calibration error defined as the percentual ratio of the difference of the forces measured in the two traps to the average of the force measured in the two traps. Data shown refer to measurements done at 13 pN of average force. Note that while the calibration error is above 15% when using the calibration factors obtained in the STOT it is reduced below 5% with the anisotropic calibration method.

Unit:pN STOT Calibration Isotropic Calibration Anisotropic Cali

µ 2.16 0.66 -0.33

σ 0.38 0.38 0.39

Table 2.10: Mean and variance for the probability distributions shown in 2.11. Both the calibration method presented in the text perform better than the original calibration. The anisotropic calibration introduces a small further improvement.

[27]. When the trap does not hit on the equilibrium position, the light exerts a force on the immobilized bead, which can be measured through the PSD (see Sec. 2). This method also allows the precise measurement of reflection effects, which are one of the sources of experimental error in the DTOT. Since one of the traps is empty, the force signal coming from the corresponding PSD is only influenced by the reflected light, which comes back along the optical path and hits the wrong PSD (see Fig. 2.2). The results concerning the shape of the trap and the effects of reflection are summarized in Figures 2.12 and 2.13. The trap has a narrow linear zone which spans the first few (' 5) pN of applied force and shows a strong non linearity from there after. The corresponding trap stiffness is plotted in the upper inset as a function of the applied force. The stiffness was obtained as a numerical derivative of the trace shown in the main figure. In both figures the reflection is shown in an inset. The data shown are compatible with an almost constant reflection, the divergence is probably due to the fact that the reflected light has a small offset. In this case the measured applied force would change sign in a region where the force signal due to the reflected light is not exactly zero. Under these conditions the ratio of the signal due to reflected light to the signal due to transmitted light would diverge. The reflection signal is below 2% when the force is above 5 pN . Problems may arise at smaller forces because there the reflected signal is comparable with the transmitted one. In section 3.2 a method to deal with this problem, at least in some special cases, is presented.

-4000 -2000 0 2000 4000

Trap Displacement (nm)

-20

-10

0

10

20

Force (pN)

Figure 2.12: Force-Displacement curve for Trap 1, obtained moving the trap with respect to the micro–pipette, the red curve is the correct force signal that gets to the right PSD, while the black curve is the force signal due to reflection on the bead surface. The reflected light has a small dependence in the trap displacement. The inset shows the relative weight of the signal due to reflection with respect to the transmitted light as a function of the applied force. The reflected signal is below 2% for forces above 5 pN. The upper inset shows the trap stiffness as a function of the applied force; the stiffness was obtained by numerical differentiation of the force-displacement data.

-2000 0 2000

Trap Displacement (nm)

-30

-20

-10

0

10

20

Force (pN)

Figure 2.13: Force-Displacement curve for Trap 2, obtained moving the trap with respect to the micro–pipette. The red curve is the correct force signal that gets to the right PSD, while the black curve is the force signal due to reflection of the light at the bead surface. The reflected light has a small dependence in the trap displacement. The leftmost lower inset shows the relative weight of the signal due to reflection with respect to the transmitted light as a function of the applied force, while the rightmost lower inset shows a comparison between the two force-extension curves of the two different traps. The reflected signal is below 2% for forces above 5 pN. The upper inset shows the trap stiffness as a function of the applied force; the stiffness was obtained by numerical differentiation of the force-displacement data.

In this chapter we discuss different equilibrium experiments performed with the DTOT introduced in Chapter 2. All the experiments discussed in this chapter replicate classical experiments. They should be seen as tests of the performance of the DTOT set–up. The first two experiments follow the lines of the seminal experiments of Meiners and Quake [5, 6] and address the hydrodynamic correlations between two untethered trapped particles and the fluctuations of two beads tethered by a dsDNA molecule. The third experiment is a hopping experiment [30], performed on a short DNA hairpin, which aims at the reconstruction of a coarse Free Energy landscape for this molecule.

3.1

The linear model

In this section we introduce a simple model which will be useful to gain insight in the physics of single–molecule experiments in a two–traps set– up. The model is simple in the sense that it is entirely linear, both in the description of the traps and tether and in the description of the hydrodynamic interactions between the beads. Even if these simplifying assumptions are not verified in most practical cases, in particular in those discussed in this thesis, the linear structure of the model allows many explicit calculations and thus the estimation of the relevance of different physical effects. The same model has been discussed several times in the literature [6, 11], but since it will play a major role in the interpretation of the experiments, I will report here some of its main features. The linear model (LM) is a schematic

• The hydrodynamic interaction between the beads, i.e. their interaction mediated by the motion of the fluid.

• The effect of thermal fluctuations.

As a first approximation, the system may be schematized as one–dimensional, considering for each bead just the degree of freedom collinear to the line which connects the centers of the traps. This direction will be denoted as ˆx. All the above listed features are subsumed in a Langevin equation of the first order:

˙

x(t) = ¯µ(x) (−∇U (x) + η(t)) , (3.1)

where x = (x1, x2) represents the position of the beads (Fig. 3.2). The potential U (x) = U1(x) + U2(x) + UM(x) is due to the contributions of the traps and of the tether, while the matrix ¯µ(x) describes both friction effects and hydrodynamic interactions. The effect of friction on the beads is of course described by the Stokes law, while the hydrodynamic interaction can in general be taken into account through the off–diagonal term in the mobility matrix (the mobility matrix itself is symmetric as a consequence of Onsager’s reciprocal relations). An approximate theory for the hydrodynamic coupling, valid when the separation between the two beads is large with respect to the bead radius, is due to Oseen [25]. In this approximation, the mobility matrix is given by: ¯ µ = 1/γ1 1/Γ 1/Γ 1/γ2  , (3.2)

where γi = 6πηlri and Γ = 4πηlR12. The last term η in (3.1) is a stochastic term due to thermal fluctuations.

In the present setting, a linear model is obtained by approximating the force-extension curves of both the traps and the tether by linear functions, expanding around the mean extension. The approximation leading to the the linear model is largely acceptable when one deals with equilibrium ex-periments, e.g. experiments in which the trap–to–trap distance does not vary in time. This is not true in non–equilibrium experiments, where the traps are moved, as in this latter case nonlinearities in the force-extension

Figure 3.1: Setting the stage for the linear model (LM). The experimental set–up composed by the optical traps and the two tethered beads will be considered as one dimensional. The motion of the beads will be projected onto the line connecting the two trap centers. The relevant variables in friction and hydrodynamic interactions are the beads’radii and the distance between the centers of the beads.

Figure 3.2: In the linear model (LM) both the traps and the molecular tether are represented as Hookean springs with spring constants k1, k2 and km respectively.

The coordinate system will have as origin the position of the first trap (XT

1) t = 0. curves both of the traps and of the biological tethers can be relevant. Other nonlinear effects may be due to hydrodynamic interactions, although these do not affect equilibrium quantities, as will be shown in Section 3.2. The linearization of Oseen’s theory is straightforward when the fluctuations in the bead-to-bead distance x2− x1 = R12 are small with respect to the mean bead-to-bead distance, as in this case Γ ' 4πηl hx2− x1i, and the mobility matrix is constant, ¯µ(x) ' ¯µ. The equation of motion in the linear model are:

˙

x(t) = ¯µ (F(t)) = ¯µ −¯k(x(t) − x0(t)) + η(t)
. (3.3) Here x(t) = (x1(t), x2(t)) are the positions of the two beads and x0(t) = (x0

1(t), x02(t)) are the mechanical equilibrium positions of the beads (which can be time-dependent in a pulling experiment).

Consider the potential (see Fig. 3.2): U (x) = 1 2k1(x1− X T 1 ) 2 +1 2k2(x2− X T 2 ) 2 +1 2km(x2 − x1) 2 . (3.4)

This can be put in normal form as: U (x) = 1 2(x − x 0_{) · ¯k(x − x}0_{) +} 1 2kef f(X T 2 − X T 1) 2_{. (3.5)} where ¯ k = k1+ km −km −km k2+ km  , (3.6) keff = k1k2km det(¯k) , (3.7)

clearly distinguished since they will play a role in the computation regard-ing fluctuation theorems. Takregard-ing the gradient of (3.5) yields the first term contributing to the force in (3.1), i.e. the deterministic and conservative con-tribution arising from the elastic interactions in the system. The second term η0 is the stochastic contribution to the force due to thermal fluctuations. We will model this contribution as a white noise:

hη(t)i = 0 (3.9)

hη(t) ⊗ η(s)i = 2kBT ¯µ−1δ(t − s). (3.10) As a first result concerning DTOTs, I discuss the variance of the force signal in the LM. The variance of the force signal is a very important quantity from the experimental point of view, because it limits the resolution of the measurements that can be performed (see Section 3.5 or ref. [11]). The force signal in the LM is the vector:

f = (f1, f2) = ¯kD(x1− X1T, x2− X2T) = (k1(x1− X1T), −k2(x2− X2T)), (3.11) with ¯ kD =  k₁ 0 0 −k2  . (3.12)

Its covariance is related to the covariance of x, the vector containing the positions of the beads by:

¯

σ_f2 = hf ⊗ f i − hf i ⊗ hf i = ¯kDσ¯x2¯kD, (3.13) Where here and in the following we denote by h·i the average with respect to the equilibrium distribution ρ(x), which in the case of the linear model is given by the Boltzmann distribution and is independent of hydrodynamic correlations,

ρ(x) = 1 Ze

−β₂(x−XT)·¯k(x−XT)_{. (3.14)}

where β is the inverse temperature and Z the normalization constant. The covariance of the beads position is proportional to the inverse of k

¯

tions of the signals coming from each trap. This is a tool we are going to repeatedly use in our analisys. The generalized force signal fφ is defined as: fφ= φf1− (1 − φ)f2 = (φ, φ − 1) · (f1, f2) = Φ · f , (3.17) with Φ the vector (φ, φ − 1). The variance of fφ is obtained from (3.16) as

σ2_φ= Φ · ¯σ_f2Φ = kBT  (k2)2(k1+ km) k2km+ k1(k2+ km) − 2k2φ + (k1+ k2)φ2  . (3.18) This is a very important result: formula (3.18) shows that from a single equi-librium measurement of the force fluctuations in a system like the one shown in Fig. 3.1 it is possible to recover the stiffness of both traps and of the molecular tether. This can be done, for example, by computing the variance of fφfor different values of φ and then fitting the result to a parabola. More-over from (3.18) it is possible to define the generalized signal that minimizes the variance of force fluctuations f_φˆ: the minimum of σφ2, which we denote σ2_ˆ

φ is obtained for φ = k2

k1+k2. Evaluating (3.18) for φ = ˆφ one obtains:

σ_φ2_ˆ = k

2 1k22

(k1+ k2)(k1(k2+ km) + k2km)

. (3.19)

In the case k1 = k2 ≡ kT, (3.19) takes the simpler form: σ_φ2_ˆ = kBT

k2 T kT + 2km

. (3.20)

This is to be compared with the variance of force fluctuations in a STOT setup. Denoting by kT the stiffness of the single trap and by km again the stiffness of the tether we have [28]

σ_s2 = kBT k2

T kT + km

. (3.21)

Comparing (3.21) and (3.20), we note that force fluctuations are larger in a STOT, for the same trap and tether stiffness conditions. The reduction of force fluctuations can in turn increase the signal to noise ratio, making this feature of the DTOT especially interesting.

11.5 12 12.5 13 13.5

Force (pN)

Probability Density (pN)

Figure 3.3: Fluctuations of the generalized force fφ (defined in the text). The

data shown are obtained in the experiments described in Section 3.4. Main figure: probability density for the force signal fφ for different values of φ. The

probabil-ity densities are obtained by convolution of a Gaussian kernel with a frequency histogram. The small shift observed in the mean of the probability density is due to calibration errors. In this case calibration errors amount to less than 1%. In-set: variance of the probability density for different values of φ together with the parabolic fit. The fit allows the reconstruction of the stiffness of the tether and of the traps.

d dt ¯ Cx(t) = h x0− x0
⊗ ˙xti = hx0⊗ −¯µ¯k(xt− x0) + η
i = = −¯µ¯k¯Cx(t). (3.23)

This differential equation must be solved with the initial condition C(0) = kBT ¯k−1, so that

¯

Cx(t) = e−¯µ¯ktkBT ¯k−1. (3.24) Although it has a simple functional form, the correlation function can have a rather complex expression. A great simplification is obtained when con-sidering a system composed by two identical traps and beads. i.e. the two traps have the same stiffness kT and the two beads have the same friction coefficient: ¯ k = km+ kT −km −km km+ kT  ¯ µ =  1/γ 1/Γ 1/Γ 1/γ  . (3.25)

In this case, and only in this case, the product ¯µ¯k is symmetric, with eigen-vectors √1 2(1, 1) 1 √ 2(1, −1) and eigenvalues: λ1 =  1 γ − 1 Γ  (2km+ kT) (3.26) λ2 =  1 γ + 1 Γ  kT (3.27)

having the dimension of an inverse time. These eigenvalues set the relaxation time of fluctuations in the linear model. For bead sizes in the micron range and stiffnesses in the range of 0.01 pN/nm, the relaxation times are in the range 1 − 10 ms. Let V be the matrix whose columns are the eigenvectors of ¯

µ¯k then the correlation function can be written as: ¯ Cx(t) = ¯VT  e−λ1t ₀ 0 e−λ2t  ¯ VkbT ¯k−1 (3.28)

A physical situation in which the correlation function has some interesting features is that of a pure hydrodynamic interaction (km = 0). In this case (3.26) becomes:

¯ Cx(t) =  e−t/τ_{cosh(t) e}−t/τ_sinh(t) e−t/τsinh(t) e−t/τcosh(t)  kbT kT . (3.30)

Since force and bead position are linearly related in this model, the propor-tionality factor being kT, the force correlation function is simply

¯ Cf(t) =  e−t/τ _{cosh(t) e}−t/τ _sinh(t) e−t/τ sinh(t) e−t/τ cosh(t)  kTkbT. (3.31) In the next section I will show how to use the correlation function to measure the strength of the hydrodynamic coupling.

3.2

Direct measurements of the hydrodynamic

coupling

Many interesting physical systems, among which colloidal suspensions, so-lutions of polymers and proteins, are affected by hydrodynamic interactions [26]. In particular the microscopic dynamics of proteins is crucially affected by the hydrodynamics of the solvent, as shown for example in [26]. The DTOT presented here allows for the direct measurement of hydrodynamic interactions between individual colloidal particles, obtained from the cross correlation of their fluctuations within the optical traps. Hydrodynamic in-teractions appear as a pronounced time–delayed dip in the cross correlation. Both the fact that the beads are anticorrelated and the fact that the effect is time delayed are not easily explained from a physical point of view: at low Reynolds numbers the dynamics is determined solely by instantaneous forces. The theoretical description of the effect is represented by the Oseen tensor which can be derived directly from the Stokes equation, the creeping flow limit (Stokes limit) of the Navier-Stokes equation. In this limit both the inertial effects of the motion of the fluid and of the particle are neglected, making the time delay in the cross correlation a non trivial effect. We studied the fluctuations of 4 µm silica beads trapped in the DTOT at different trap– to–trap distance in a PBS buffer. Salt conditions (1 M NaCl) give a Debye

less, in absence of a tether this effect is easily removed, since hydrodynamic couplings do not affect the equilibrium properties, but only time correlations. To better understand this fact one should think to the simpler case of over-damped diffusion in a quadratic potential well: in this case it is the friction coefficient, one–dimensional analog of the hydrodynamic coupling, that does not influence the stationary fluctuations. The size of equilibrium fluctuations is completely determined by the temperature and the stiffness of the well. As a consequence the variance and covariance of the beads fluctuations in the trap should be the same whether hydrodynamic couplings are present or not. In particular, the covariance of the force signals should be zero. The effect of reflection on fluctuation measurements appears as a spurious non–zero co-variance in the force signals coming from each trap. This can be understood in a simple way: by reflection some of the signal due to trap 1 is read as coming from trap 2 and vice versa so that we can assume that the real signal f = (f1, f2) is disturbed by reflection in the following way:

¯

f1 = f1+ af2 (3.32)

¯

f2 = f2+ af1, (3.33)

with a( 1) the small fraction of reflected light. Here f1 and f2 have zero covariance,

hf1f2i = hf1ihf2i;

the covariance matrix of the signal ¯f will then be, to the first order in a: ¯ ω_f2 =  σ2 11 a(σ112 + σ222 ) a(σ2₁₁+ σ2₂₂) σ₂₂2  , (3.34) where σ2

ii = h(fi2i − hfii2i. Reflection effects can be removed through a rotation ¯R that diagonalizes the covariance matrix:

¯

σ_f2 = ¯R¯ω_f2R¯−1, (3.35) where ω2_f denotes the covariance matrix in presence of reflection effects and σ2

f represents the corrected covariance matrix. Applying the same rotation to the force trace yields the corrected signal: