Force Measurements with Optical Tweezers

Force Measurements with Optical Tweezers

Othmar Marti Institut für Experimentelle Physik Universitaet Ulm Albert-Einstein-Allee 11 D-89069 Ulm, GERMANY

Katrin Huebner Institut für Experimentelle Physik Universitaet Ulm Albert-Einstein-Allee 11 D-89069 Ulm, GERMANY

Handbook of Nanotechnology, (B. Bhushan, ed.) (3rd Edition) Chapter D.32, pages 1013-1022 For publication in Nanotechnology Handbook (B. Bhushan, ed.), Page proofs available Springer-Verlag, Heidelberg, Germany

1

Abstract

An optical tweezer is a scientific instrument that uses a focused laser beam to provide an attractive or repulsive force, depending on the index mismatch, to physically hold and move microscopic dielectric objects. “Since their invention just over 20 years ago, optical traps have emerged as a powerful tool with broad-reaching applications in biology and physics. Capabilities have evolved from simple manipulation to the application of calibrated forces on - and the measurement of nanometer-level displacements of - optically trapped objects.”[Neuman2004]. The ability to apply forces in the pico-Newton range to micron-sized particles while simultaneously measuring displacement with nanometer resolution is now routinely adopted to the study of molecular motors at the single-molecule level [Becker2005], the physics of colloids and mesoscopic systems [Bar-Ziv1997,Clapp2001], and the mechanical properties of polymers and biopolymers [Grier2003,Svoboda1994,Kuyper2002]. In parallel with the widespread use of optical trapping, theoretical and experimental work on fundamental aspects of optical trapping is being actively pursued [Visscher1992a,Visscher1992b,Mazolli2003]. In this chapter we will give a short overview of trapping and detection principles, different calibration methods, as well as the influence of surfaces and viscosity will be discussed. In the end there will be given a short insight in the application of optical tweezers in cell biology.

2

1. Optical tweezers 1.1.

Principles of optical trapping

1.2.

Detection principles

1.3.

Photonic Force Microscope

1.4.

Position calibration

1.5.

Force calibration

2. Influence of surfaces and viscosity 3. Thermal noise imaging 4. Applications in cell biology 4.1. Applications to the cytoskeleton

3

1.

Optical tweezers

Although James Clerk Maxwell in 1873 already showed theoretically that light can exert optical force, the experimental proof could not be given until the advent of lasers in 1960. In 1970 Ashkin was the first, who succeeded in accelerating and trapping micron-sized particles using the force of radiation pressure from a continuous laser [Ashkin1970]. Sixteen years later Ashkin and co-workers demonstrated the first single-beam gradient force optical trap [Ashkin1986]. From that time on such setups, using a single, highly focused laser beam to trap small particles in three dimensions were called "Optical Tweezers"(OT). Nowadays OTs have become a powerful tool with many applications in physics and biology [Svoboda1994]. They have been used to trap dielectric spheres, atoms [Ashkin1978], viruses, bacteria, living cells [Lim2003], organelles [Ashkin1989], small metal particles, and even strands of DNA.

1.1. Principles of optical trapping Basically an Optical Tweezer consists of a trapping laser (with wavelength λ), an objective lens with high numerical aperture (NA) to focus the laser beam, a detection lens and a quadrant photodiode for detecting the scattered and unscattered light (see Fig. 1). Due to the high NA of the objective strong intensity gradients in three dimensions arise. A dielectric particle near the focus experiences a force in the direction of the light gradient, called gradient force (Fig. 2), as well as one in the direction of light propagation, called scattering force. Both forces originate from the change of momentum of the photons, which causes, according to Newtons third law, also a change of momentum of the bead. In Figure 2 the optical path of exemplary beams and the

4

resulting gradient forces are shown in dependency of the light intensity profile. The scattering force Fscat arises from absorption and reflection of the incident photons (the so called radiation pressure) and is the dominant force in most conventional cases. Only if there is a steep intensity gradient in the beam, the gradient force Fgrad can compensate Fscat. The balance between the scattering force and the gradient force in axial direction results in a stable trapping slightly behind the focus. For small displacements of the particle (≤ λ/2 in axial and λ/4 in lateral direction) the resulting, restoring optical force Fopt= Fscat + Fgrad

(1)

is linear and the trap acts like a Hookean spring in three dimensions. For x,y and z direction follows: Fx=kx*Δx

Fy=ky*Δy

Fz=kz*Δz,

(2)

where Δx,y,z is the displacement and kx,y,z the characteristic trap stiffness in the particular direction. The traps stiffness grows linear with laser intensity. As the stiffness in axial (z-) direction is smaller than in the x,y plane the trapping volume forms an ellipsoid. To estimate the optical forces acting on a trapped particle theoretically some different approaches have to be made: (1) When the trapped particle is much larger than the wavelength of the laser, conditions for Mie scattering are satisfied and the forces can be calculated by simple ray optics (Fig.2). For detailed informations and calculations see e.g. Ashkin1992, Visscher1992b. (2) When the trapped particle is much smaller than the wavelength Rayleigh scattering has to be applied. Therefor Harada and Asakura approximated the sphere as an simple dipole, see [Harada 1996]. Also Visscher and Brakenhoff did a theoretical disquisition on forces on spherical particles

5

in an optical trap in the Rayleigh regime, using electromagnetic diffraction theory [Visscher1992a]. For many biological applications the laser wavelength and the particle size are in the same range, so none of the extreme cases mentioned above is reliable. Approaches for a more generalized electrodynamic theory were made by Barton and co-workers [Barton1989]. Also Rohrbach and Stelzer did some work on the theoretical description and simulation of forces in optical traps [Rohrbach2001] using electromagnetic theory. Rohrbach also showed the agreement of his theoretical estimations with experimental results for particles in the range of the wavelength or slightly beneath [Rohrbach2005].

1.2. Detection principles Because of the linear relation between force and displacement, the force acting on the trapped particle can be measured by the displacement from the resting position. Pralle and co-workers [Pralle1999] introduced a theory to get the full three-dimensional position information of the particle within the trapping volume by detecting the interference of the scattered and unscattered light with a four quadrant photodiode (QPD). The position in the x,yplane can be obtained by subtracting the left and right quadrant signals (Left-Right) and the lower and upper quadrant signals (Top-Bottom), this was already known. The axial displacement from the rest position can be detected as a change in the sum signal of the QPD. This is due to the phase difference between the scattered and the unscattered light based on the Gouy Phase shift (phase anomaly)[Born]. Which is a continuous shift of the phase over the focal region inherent in every non plane wave. Rohrbach and Stelzer did a more exact description of the three-

6

dimensional position detection for arbitrary particle sizes and investigated theoretically the accuracy and sensitivity of the detection system for various sphere parameters (size, refractive index) [Rohrbach2002].

1.3. Photonic Force Microscope The possibility to detect the position of a trapped bead in three dimensions with very high temporal (µs) and spatial (nm) resolution makes it feasible to analyze the probes Brownian motion within the trapping potential. Such a setup is often called Photonic Force Microscope (PFM). As the thermal fluctuations of small particles are determined by their environment, the PFM enables the measurement of physical parameters such as viscosity [Pesce2005,Pralle1998], diffusion, temperature and small forces in the local environment of the probe.

1.4. Position Calibration A frequently used technique to calibrate the position detector is the so-called attached-bead method. Following this method the position detector response is recorded, meanwhile a bead fixed to the surface is moved through the laser focus by a known length within the linear response of the detector. Unfortunately this method is affected by significant bias caused by proximity surface effects and by spherical aberrations. Another drawback is caused by the axial dependence of the lateral position signals. So it is necessary to precisely match the axial position of the tethered bead to that of the trapped bead, which is not that easy. An other possibility is the position calibration based on thermal motion, for example using the power spectral density as described for force calibration below.

7

An other problem using oil-immersion objectives is, that the calibration factor for the position detection decreases as the focal plane penetrates in the sample, due to spherical aberrations mainly caused by refractive index mismatch [Pesce2005]. Using water-immersed objectives this effect was not noticed.

1.5. Force Calibration To get quantitative data it is necessary to do an accurate force calibration. For these purposes there are several methods, which can be divided into two main groups. On the one hand calibration against known forces, usually the viscous drag force, on the other hand methods using Brownian motion and statistics.

a) Calibration against viscous drag In this cases a viscous drag, generated by a relative motion of the particle with respect to the surrounding liquid is compared with the optical trapping force. This can be done by translating the laser spot with the trapped particle, by using a static laser trap and translating the microscope object slide or by directly generating a liquid flow. According to Stokes law the friction force can be written as follows: Ffrict= 6πηav

(3)

Where η is the viscosity, a the radius of the sphere and v the velocity. A precise measurement of the flow velocity can be obtained by trapping a bead in an optical trap without flow and then oscillating the bead against the surrounding liquid at a fixed frequency and amplitude. According to [Williams], for a sinusoidal oscillation the velocity is given by: v = ωx0 cos(ωt)

8

(4)

Ffrict= 6πηaωx0 cos(ωt)

It follows:

(5)

In this case all of the quantities on the right of Eq.5 are known, so we know the applied force as a function of time. The measured signal S as a function of time will be S = Aω cos(ωt)

(6)

One can then measure this signal at several frequencies, determine the amplitude Aω according to Eq. 6 and fit the resulting data to a straight line to obtain a value for A. Substituting Eq. 6 into Eq. 5 gives F = (6πηax0 /A)S ≡ D·S

(7)

with the calibration factor D. Whenever a signal S is detected with a bead in the trap, one can directly calculate the force on the bead using Eq. 7. Although apparently simple, this calibration method is complicated by the dependence of the viscosity on the temperature and the hydrodynamic corrections which become necessary when the distance to the cover slip is of the order of the bead diameter [Happel]. Taking into account errors due to bead diameter, temperature, detector calibration and statistics, in [Florin1998] the absolute error of this method was about 20%. This relative large error and the restriction to lateral dimensions are disadvantages for this method.

b) Statistical methods

This methods are based on the thermal fluctuations of the particle within the trapping potential. One advantage of all these methods is, that they also can be used to calibrate the trap in axial direction.

9

Again there are different approaches:

Equipartition theorem/Mean square displacement As the trapped particle is in equilibrium with its surrounding and assuming a harmonic trapping potential for every direction the equipartition theorem says: ½ ktrap = ½ kBT

(8)

With the trapping constant ktrap, the Boltzmann constant kB, the temperature T and the mean square displacement from the resting position . So by measuring the positional variance of the trapped bead, ktrap can be determined. One advantage of this method is, that it does not explicitly depend on the viscosity of the surrounding liquid, nor the beads shape or hight above the surface. For this method [Florin1998] calculated an absolute error of 7%. Care has to be taken with this method, because any added noise and drift in position measurements serve in an increase of the variance and therefore in a decrease of the estimated trap stiffness.

Power spectrum/ Corner frequency The particles motion can be described by the Langevin equation: Ffric(x,t) +Fopt(x) = Ftherm(t) ↔ γ0v(t)+ ktrapx(t) = (2 kBTγ0)½η(t)

(9)

with the friction coefficient γ0, x(t) the trajectory of the Brownian motion. After dropping inertial terms and the adoption of D = kBT/γ0 and the corner frequency fc = ktrap/2π γ0 one gets the power spectral density (PSD) of the mean square displacement of a over damped oscillator, which is expected to be Lorentzian, see e.g. [Berg-Sørensen]. PSD = D/{(2π²)( fc²+f²)}

(10)

So by fitting the Lorentzian to the PSD one gets the corner frequency and hence the force constant ktrap. An advantage of this method is, that the detector calibration need not be known. 10

Florin calculated a total error of about 11% for this method. In [Berg-Sørensen] the measurement and accurate fitting of PSDs is discussed in detail. Using this method, the viscosity of the surrounding liquid has to be known.

Autocorrelation Function (ACF) The ACF of the position distribution is given by: =r-2exp(- t/τ)=r-2exp(- tktrap/γ)

(11)

with the autocorrelation time τ = ktrap/γ. So by fitting this exponential function to the ACF, one can calculate the trap stiffness ktrap , when the friction coefficient γ, thus the viscosity, is known.

Boltzmann statistics Knowing the position distribution of the trapped particles motion, one can calculate the trapping potential (see section 3). By fitting a harmonic potential one easily can calculate the force constant. As this method is closely related to the “mean square displacement” in [Florin1998] also a total error of about 7% is determined. Combining the Boltzmann statistics with other methods (e.g. corner frequency) allows the determination of additional parameters, like the local viscosity, which was not accessible before [Florin1998]. This is very useful, because the viscosity is one of the most uncertain parameters, as it depends on both, the temperature and hydrodynamics.

The mentioned statistical calibration methods can be used in lateral as well as in axial direction. One important point is also, that they can be used in situ and calibration can be done in the region of interest. This especially is important for measurements near surfaces, because there the viscous drag differs from that in bulk (as we will see later more detailed).

11

In [Tolic2006] an approach combining corner frequency and drag force measurements is introduced. Neither the viscosity, nor the size of the trapped object, nor its distance to nearby surfaces needs to be known and it also can be applied in situ in all spatial dimensions. A further advantage of combining these methods is the possibility to get both the force calibration as well as the position calibration at the same time and in situ. Recently Fischer and Berg-Sørensen [Fischer2007] implemented a calibration method, which also can be used in viscoelastic media, such as cells or polymer gels. This is, especially for biophysical applications a big improvement. Their method can be applied to general viscoelastic media and also the size and shape of the trapped particle need not be known. They combined a passive and an active measurement of Brownian motion and calculated the friction relaxation spectrum. In the passive part stage or laser stays undriven, in the active part stage or laser is driven sinusoidally. To combine both parts the main assumption is, that the friction relaxation spectrum of the driven and undriven system is the same for small disturbances. This adoption is motivated through Onsager’s regression hypothesis [Onsager1931] which is a consequence of the fluctuation– dissipation theorem, states that ‘the regression of microscopic thermal fluctuations at equilibrium follows the macroscopic law of relaxation of small non-equilibrium disturbances’ [Onsager1953]. In [Fischer2007] an instruction how to calibrate in this way is given and they showed by a simulation that the method seems to perform well.

2.

Influence of surfaces and viscosity

Particles confined in an optical trap behave as local probes to explore the surrounding medium, so they can be used to measure the local viscosity. To determine the local viscosity by means of OTs or PFMs there are mainly three unknown quantities: (1) the traps stiffness,(2) the calibration 12

factor of the position detector and finally (3) the viscosity. As mentioned above, by combining calibration against viscous drag with a statistical method one can determine these three variables at one time. Using two thermal analysis techniques instead, viscous drag or position calibration factor need to be known. For a rheological application the calibration factor should be determined independently. As the attached-bead method includes some sources of error, Pesce and Sasso [Pesce2005] described a calibration procedure, which is based on a comparison of two independent and simultaneous measurements of the trapped bead displacements: one is obtained from an image analysis by means of a CCD camera while the second one is derived from the signal of a quadrant photodiode used as a position detector. Afterwards they calculated the trap stiffness and the viscosity using the PSD. So viscosity of Newtonian fluids can be measured very accurate, as the calibration of the system (position and force) can be done in the region of interest. Pesce also predicted that this method can be extended also to more complex fluids, like polymeric solutions, gels or colloids to investigate their viscoelastic response. Another approach to analyze viscoelastic properties is given in [Fischer2007], as described above. As already mentioned the influence of surfaces cannot be neglected. Stokes law only holds for beads in bulk solution. However local viscosity of a fluid depends not only on parameters such as the chemical composition and temperature, but also changes, owing to spatial constraints. Particles close to a surface are partially confined and hence the particle’s diffusion coefficient D is reduced. Pralle and co-workers [Pralle1998] investigated the effect of sphere-surface separation to the local viscosity. Therefor they combined Boltzmann statistics to calibrate the force with investigating the autocorrelation function (this is closely related to the PSD, as the PSD is the Fourier transformation of the autocorrelation function). They showed, that the Diffusion constant (D = kBT/(6πηa)), and therefore the local viscosity remains constant until the distance between bead and surface is in the order of the spheres radius. For smaller distances the diffusion constant

13

decreases, while the viscosity increases. The results are in good agreement with theoretical estimations made in [Brenner]. Also Clapp and Dickinson [Clapp2001] investigated the interaction between particles and surface. They also observed light interference effects when the trap focus is near the solid-liquid interface, which makes it difficult to measure in this region.

3.

Thermal noise imaging

In the presence of an external potential, originating for example from external structures or molecules tethered to the bead, the trajectory of the fluctuating bead in the trap alters, see Fig.3. The idea is, that by subtracting the measured distorted potential from the trapping potential in bulk solution one gets the external interaction potential [Rohrbach2004]. And therefore one can calculate the interaction force and get information about parameters like elasticity for example. Following Rohrbach and co-workers [Rohrbach2004], the Langevin equation for a bead trapped in an PFM can be written as: Ffric(r,t)-Ftherm(t)+Fopt(r)+Fext(r,t) = 0

(12)

With the optical force due to the trap like in equations (1,2), the friction force:

Ffric(r,t)=γ(r)v(t)

(13)

with the viscous drag γ(r)=6πaη(r), where a is the probes radius and η(r) the local viscosity, a random, thermal force Ftherm(t), which depends on temperature and viscosity and eventually an external force Fext(r,t).

14

Using Boltzmann statistics one can show, that there is a simple connection between the potential W(r) and the normalized position distribution (or probability density) p(r), given by W(r)=-ln(p(r)) [Rohrbach2004]. In Fig.4 exemplary the position distribution of a trapped 1µm polystyrene bead, and the calculated potential in z-direction is shown. As described in [Tischer2001] or [Rohrbach2004] the photonic force microscope can be used as an imaging device by scanning the optical trap and thus the probe across a sample surface. The probe fluctuations are recorded with nanometer spatial and microsecond temporal resolution, and the scanned objects are reconstructed from three-dimensional position histograms of the position fluctuations.

4.

Applications in cell biology

There is a wide range of applications for optical tweezers or photonic force microscopes in biophysics and cell biology. There are already some reviews dealing with biophysical applications [Svoboda1994, Gross2003, Kuo2001, Kuyper2002]. On the one hand cells [Lim2003] or vesicles within cells [Ashkin1989] can be trapped themselves. On the other hand one can use small dielectric beads as a force transducers. For cell biology it is necessary that measurements can be done in liquid, furthermore often forces, exerted with an atomic force microscope are too strong and lead to a damage of the cells or tissues. Other advantages of using OT are [Kuyper2002], (1) that they easily can be integrated into microscope imaging systems and offer a sterile and noninvasive instrument to manipulate biological particles ranging from tens of nanometers to many micrometers, (2) in comparison with

15

other techniques, such as glass micropipettes, optical tweezers offer a more versatile and facile method for micromanipulation, (3) optical tweezers offer excellent spatial resolution and dexterity in micromanipulation with small forces (pN) and (4) the near-infrared wavelength (800 – 1064 nm) used in most optical traps produces rather minor effects, if any, on the function of biological particles and the viability of cells.

4.1. Applications to the cytoskeleton One important application in cell biology is to determine the mechanical properties of cells, respectively their cytoskeleton. As cells are highly dynamic, they exert and respond to forces in their environment, this behavior is closely related to the mechanical properties of the cells. So, to understand the cells behavior, e.g. motility, differentiation etc., knowledge of cell mechanics is essential. The viscoelastic behavior of cells is mainly determined by their cytoskeleton. The cytoskeleton is a three-dimensional, heterogeneous and dynamic network, consisting of three major biopolymer classes: filamentous actin (F-actin), intermediate filaments (Ifs) and microtubules (MT). One focus lies on the IFs, because they form a scaffold defining shape and mechanical properties of cells [Herrmann2003]. Already some work was done on the keratin 818 Ifs of panc-1 cells (cells from pancreatic cancer), as the healing rate of this cancer type is very bad (only about 3%). In [Beil2003] it is shown, that sphingosylphosphorylcholine regulates keratin network architecture and visco-elastic properties of human cancer cells. To get more informations about the viscoelastic properties of the keratin network, we perform photonic force microscope measurements to the extracted cytoskeleton. Extraction is done as described in [Svitkina1998] but stopped before fixation and drying. So it is supposed that only the keratin network, including the nucleus, remains. A electron microscopy picture of the then fixed and dried network is shown in Fig.5. In Fig.6 the 2D histograms of the particles position near the 16

cytoskeleton are shown. As one can see, there are obvious differences to the elliptical shape in bulk. In y-direction there are two places where the particle is more often, so between these two places, there must be some disturbance, e.g. parts of the cytoskeleton. As described in Sec.3 by scanning over the cytoskeleton point by point and combining the resulting histograms one can get an idea of the surface topography. It seems possible to put beads of different sizes also into the cytoskeleton, by “feeding” the cells during cultivation with the beads. So this technique is also promising to study the form and mesh size of the network from inside, by investigating the internal position distributions and the confined diffusion of the particles inside. A first measurement of the potential (here in axial direction) of a bead close to the cytoskeleton is shown in Fig.7. The broader potential including a harmonic fit is the one of a bead in bulk, the smaller, somehow asymmetric one, is the potential measured when the bead pushes against the cytoskeleton. We made a first approximation to determine the Young's modulus by a fit to the potential. To model this potential energy, we assume that the cytoskeleton is a continuous body. This is justified, as long as the characteristic mesh size of the cytoskeleton is much smaller than the diameter of the sphere R. Then the contact is of the Hertz type. The force in this type of contacts is related to the radius a of the sphere and the effective modulus K [Gigler]:

F ( z ) = K · a·z 3

(14)

The potential energy is obtained by integration.

2 Ecyto ( z ) = K ·  a·z 5 5

(15)

By adding the trapping potential one gets the total obtained potential:

E= Etrap ( z ) + Ecyto ( z − z= tot 0)

1 5 2 ktrap ·z 2 + K ·  a·( z − z0 ) 2 5

with a possible offset z0 for z ≥ z0.

17

(16)

Assuming an isotropic material and a Poisson number of 0.5 , we get: K=

8 E 9

(17)

with the Young's modulus E. In Fig.8 on example of the potential and the according fit is shown (the fit was done with Sigmaplot 9.0). Here we obtained:

E = 34 Pa.

Other measurements delivered moduli in the same range. More measurements to have a statistic and comparing the moduli at different places over the cell (in the surrounding of the cell or near the nucleus) are in work. As discussed in sections 1.5 and 2 one approach to investigate the viscoelastic properties of the surrounding of the bead is to have a closer look to the ACF. In Fig.9 the ACF of a bead in bulk and one near the cytoskeleton is shown. The according autocorrelation times are obtained by fitting Eq.11 to the data. We get τbulk= 0,59 ms and τcyto= 0,69 ms. We are planning to combine the measurements of the ACF with other calibration methods as described in Sec. 1 and 2 to get quantitative information of the local viscoelastic properties. Altogether PFM seems to be a very promising technique to investigate the cytoskeleton and its mechanical, especially viscoelastic properties. So changes of the cytoskeleton after adding drugs or special proteins can be obtained. Further, by multiple traps or by combining the PFM with a high speed camera the correlation of the Brownian motion of two or more particles within the network can give information about force transfer within the cytoskeleton.

[Ashkin1970] ‘‘Acceleration and trapping of particles by radiation pressure,’’A. Ashkin, Phys. Rev. Lett. 24 (4), 156–159 (1970)

18

[Ashkin1978]‘‘Trapping of atoms by resonance radiation pressure,’’ A. Ashkin, Phys. Rev. Lett. 40 (12), 729–732 (1978). [Ashkin1986]‘‘Observation of a Single-Beam Gradient Force Optical Trap for Dielectric Particles,’’ A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, Opt. Lett. 11 (5), 288–290 (1986) [Ashkin1989]”Internal cell manipulation using infrared laser traps”, A. Ashkin, J. M. Dziedzic, Proc. Nadl. Acad. Sci. USA Vol. 86, pp. 7914-7918, (1989) Cell Biology [Ashkin1992]‘‘Forces of a Single-Beam Gradient Laser Trap on a Dielectric Sphere in the Ray Optics Regime,’’ A. Ashkin, Biophys. J. 61 (2), 569–582 (1992) [Barton1989]‘‘Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,’’ J. P. Barton, D. R. Alexander, and S. A. Schaub, J. Appl. Phys. 66 (10), 4594–4602 (1989). [Bar-Ziv1997] “Localized Dynamic Light Scattering: Probing Single Particle Dynamics at the Nanoscale”, R. Bar-Ziv, A. Meller,T. Tlusty, E. Moses,J. Stavans, S. A. Safran, Phys. Rev Let. 78 (1) 154-157 (1997) [Becker2005] “Three-dimensional bead position histograms reveal single-molecule nanomechanics”,N. B. Becker,S. M. Altmann, T. Scholz, J. K. H.Hörber, E. H. K. Stelzer,and A. Rohrbach, Phys. Rev. E. 71, 021907 (2005) [Beil2003] “Sphingosylphosphorylcholine regulates keratin network architecture and visco-elastic properties of human cancer cells”, Michael Beil,Alexandre Micoulet, Götz von Wichert, Stephan Paschke, Paul Walther, M. Bishr Omary, Paul P.Van Veldhoven,Ulrike Gern, Elke Wolff-Hieber, Juliane Eggermann, Johannes Waltenberger, Guido Adler, Joachim Spatz and Thomas Seufferlein, Nature cell Biology, Vol 5 (9), 803-811 (2003) [Berg-Sørensen2004] “Power spectrum analysis for optical tweezers”, K. Berg-Sørensen, H. Flyvbjerg, Rev.Sci.Inst. 75 (3),594-612 (2004) [Born] “Principles of optics, M.Born, E.Wolf, University Press New York, 1999 [Brenner] H.Brenner, Chem. Eng. Sci. 16 242 (1961) [Clapp2001]‘‘Three-dimensional optical trapping and evanescent wave light scattering for direct measurement of long range forces between a colloidal particle and a surface,’’A. R. Clapp, A. G. Ruta, and R. B. Dickinson,Rev. Sci. Instrum. 70 (6), 2627–2636 (1999) [Fischer2007] “Calibration of trapping force and response function of optical tweezers in viscoelastic media”, M. Fischer, K. Berg-Sørensen, J.Opt. A:Pure Appl.Opt. 9, 239-250 (2007)

19

[Florin1998] “Photonic force microscope calibration by thermal noise analysis” E.-L. Florin, A. Pralle, E.H.K. Stelzer, J.K.H. Hörber, Appl.Phys.A 66, 75-78 (1998) [Gigler] Alexander Gigler and Othmar Marti, Nanotechnology, chapter DPFM, page xx. 2007 [Grier2003] “A Revolution in Optical Manipulation”, D.G. Grier [Gross2003] “Application of Optical Traps In Vivo“, S.P.Gross, Methods in Enzymology, Vol. 361, PP. 162-74 (2003) [Happel] “Low Reynolds Number Hydrodynamic”, J.Happel, H.Brenner, (Prentice Hall, Englewood Cliffs NJ 1965) [Harada 1996]‘‘Radiation forces on a dielectric sphere in the Rayleigh scattering regime,’’ Y. Harada and T. Asakura, Opt. Commun. 124 (5–6), 529– 541 (1996 [Herrmann2003] “Functional complexity of intermediate filament cytoskeletons: From structure to assembly to gene ablation”, H. Herrmann, M. Hesse, M. Reichenzeller, U. Aebi, and T.M. Magin, Int. Rev. Cytol. 223, 83–175 (2003) [Kuo2001]‘‘Using optics to measure biological forces and mechanics,’’ S. C. Kuo, Traffic 2 (11), 757–763 (2001) [Kuyper2002] “Optical Trapping: A Versatile Technique for Biomanipulation”, C.L.Kuyper and D.T. Chiu, Appl. Spect. Vol. 56 (11), 300A-312A (2002) [Lim2003] “Large deformations of living cells using laser traps” C.T. Lim, M. Dao, S. Suresh, C.H. Sow, K.T. Chew, Acta Materialia 52 ,1837–1845(2004) [Mazolli2003] “Theory of trapping forces in optical tweezers”, A. Mazolli, P. A. Maia Neto and H. M. Nussenzveig, Proc. R. Soc. Lond. A 459, 3021–3041(2003) [Neuman2004] ” Optical trapping”, Keir C. Neuman and Steven M. Block, Rev. o. Scientific Instruments, Vol. 75, 9 pages: 2787-2809 (2004) [Onsager1931] “Reciprocal relations in irreversible processes 1”, L.Onsager, Phys. Rev. Vol 37, 405-426, (1931) [Onsager1953] “Fluctuations and Irreversible Processes”, L.Onsager, S.Machlup, Phys.Rev. Vol.91 (6) 1505-1512 (1953) [Pesce2005] “Viscosity measurements on micron-size scale using optical tweezers”, G.Pesce, A. Sasso, S. Fusco, Rev. Sci. Inst. 76 115105 (2005) [Pralle1998] “Local viscosity probed by photonic force microscopy”, A. Pralle, E.-L. Florin, E.H.K. Stelzer and J.K.H. Hörber, Appl. Phys. A 66, 71-73 (1998) 20

[Pralle1999] “Three-Dimensional High-Resolution Particle Tracking for Optical Tweezers by Forward Scattered Light” A. Pralle, M. Prummer, E.-L. Florin, E.H.K. Stelzer and J.K.H. Hörber, Micrsocopy Research and Technique 44, 378-386 (1999) [Rohrbach2001] “Photonic Force Microscopy: Simulation of principles and applications”, A. Rohrbach, E.-L. Florin, E.H.K. Stelzer, Proc.SPIE Vol.4431, 75-86 (2001) [Rohrbach2002] “Three-dimensional position detection of optically trapped dielectric particles”, A. Rohrbach, E.H.K. Stelzer, J.Appl. Phys. 91 (8), 5474-5488 (2002) [Rohrbach2004] “Trapping and tracking a local probe with photonic force microscope”, A. Rohrbach, C. Tischer, D. Neumayer, E.-L. Florin, E.H.K. Stelzer, Rev. Sci. Inst.75 (6), 21972210, (2004) [Rohrbach2005] “Stiffness of Optical Traps: Quantitative Agreement between Experiment and Electromagnetic Theory”, A. Rohrbach, Phys. Rev. Let. 95 168102 (2005) [Svitkina1998] “Correlative Light and Electron Microscopy of the Cytoskeleton”, T.M. Svitkina, G.G.Borisy, Cell Biology Chapter 28, 277-286 (1998) [Svoboda1994] ‘‘Biological applications of optical forces,’’ K. Svoboda and S. M. Block, Annu. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994) [Tischer2001] “Three-dimensional thermal noise imaging”, C.Tischer, S.Altmann, S.Fisinger, J.K.H. Hörber, E.H.K.Stelzer, E.L. Florin, Appl. Phys. Let. Vol.79 (23), 3878-3880 (2001) [Tolic2006] “Calibration of optical tweezers with positional detection in the back focal plane”, S.F.Tolic, E.Schäffer, J. Howard, F.S.Pavone, F.Jülicher, H. Flyvbjerg, Rev. Sci. Inst. Vol.77, 103101 (2006) [Visscher1992a]‘‘Theoretical study of optically induced forces on spherical particles in a single beam trap I: Rayleigh scatterers,’’ K. Visscher and G. J. Brakenhoff, Optik 89 (2), 174–180 (1992) [Visscher1992b]‘‘Theoretical study of optically induced forces on spherical particles in a single beam trap II: Mie scatterers,’’ K. Visscher and G. J. Brakenhoff, Optik 90 (2), 57–60 (1992) [Williams] “Optical Tweezers: Measuring Piconewton Forces”, M.C.Williams,

21

Figure 1

Figure 2

Figure 3

Figure 5

Figure 6