Towards Measuring Piconewton Forces with Optical Tweezers

Transcription

1 University of Ljubljana Faculty of Mathematics and Physics Seminar II Towards Measuring Piconewton Forces with Optical Tweezers Author: Matjaž Žganec Advisor: doc. dr. Igor Poberaj March 28, 2007 Abstract Optical tweezers are instruments which use laser light radiation pressure to trap micrometer-sized particles. The forces that can be exerted on trapped particles vary from 1 up to 100 piconewtons (pn), which gives way for manipulation of particles ranging from colloids to macromolecules. Although initially used for manipulating artificially manufactured beads, the instrument shows its true capabilities when applied to the field of microbiological studies. This seminar provides an explanation from a physical point of view of how an optical trap works and compares two most common optical tweezers construction designs. Then, two frequently used calibration methods for determining trap stiffness are described. Finally, the variety of possible applications where optical tweezers can come handy is illustrated by two case studies listed at the end of the seminar.

2 Contents 1 Introduction 2 2 History and Development 3 3 Optical Trapping Optical trapping theory Ray optics regime Rayleigh regime Basic Design General description Microscope Laser Classification by number of beams Force Measurement Calibration Viscous drag force calibration Brownian motion calibration Conversion to physical units Applications Studying molecular motors Manipulating small objects with holographic traps Conclusion 15 8 Acknowledgments 16

3 1 Introduction "We ve been able to achieve a resolution of three angstroms the width of three hydrogen atoms in our measurements of the progress of this enzyme along DNA. In so doing, we ve been able to visualize a backtracking motion of just five bases that accompanies RNA polymerase error-correction or proofreading." Steven Block, 2003 Although discovered by chance less than four decades ago 1, optical tweezers have quickly proven useful for controlling micrometer- and submicrometer-sized dielectric particles, ranging from colloids to atoms in extension 2. The instrument can exert a range of extremely small forces which are difficult to produce otherwise and are particularly applicable in microbiological studies. This seminar presents the physical background of how an optical trap works. Analytical derivation is possible in two limits of the ratio between the trapped particle diameter and the wavelength of the laser light used. For small particles an electromagnetic field theory is applied, whereas for large particles a simple ray treatment suffices. Besides the equations for the scattering and gradient force magnitude, a qualitative explanation of why the restoring force always points to the focus of the laser light is provided. After the basics of optical trapping are covered, an simplified scheme of optical tweezers is given, omitting all technical details about the considerations before actually building one. Two most common designs of the instrument regarding the number of laser beams used are briefly explained and pros and cons of each design are listed. Since none of the limits for calculating the forces of laser radiation on the trapped particle usually applies in real experiments, two frequently used calibration methods for determining trap stiffness are provided. The first one is based on the viscous drag force of the surrounding medium on the trapped particle. The other method involves monitoring the position of the trapped particle over time due to Brownian motion. By means of statistical analysis the trap potential and thus the force field can be calculated from the acquired data. The seminar is concluded with two short presentations of actual experiments that can be done by employing optical tweezers. The first case study shows a speed/force profile of DNA replication, whereas the second one demonstrates optical tweezers applicability on a much larger scale, such as particle arranging and sorting techniques. As an extension to simple optical tweezers, a short explanation of multiple-trap holographic system is given. 1 First observed in 1970 by Arthur Ashkin, Bell Laboratories. 2 Steven Chu, Stanford, winner of the Nobel Prize in Physics for research in laser cooling and trapping of atoms in

4 2 History and Development "Kinesin is one of the most efficient engines anyone has ever seen. Some estimates put it at near 100 percent efficiency. It s an amazing little thing." Joshua Shaevitz, 2003 The optical trapping phenomenon was first reported by Arthur Ashkin in 1970 (see [1]), when he used a focused laser light to trap small spheres immersed in water. Since the radiation pressure was too large, the particles started collecting at the chamber wall. By using an additional oppositely directed laser beam, Ashkin constructed an optical trap which was capable of holding a particle in three dimensions 3. Ten years later, the same principles were applied to microbiological sciences, when his team successfully trapped individual tobacco mosaic virus and Escherichia Coli bacterium (see [2]). Later on, other researchers such as Steven Block, used optical traps on a molecular scale to measure the forces biomolecular motors are capable of producing. One of such studies is presented at the end of the seminar. The principles of optical trapping served as a foundation to construct a device which can cool and trap atoms. This research earned Chu the Nobel Prize in Physics in With this extension, the trapped particle diameter scaled from several micrometers down to subnanometer range. Recent advances in constructing optical tweezers have paved way for new practical applications, such as cell sorting which is presented in a case study at the end, probing cytoskeleton (see [3]), and studying cell motility. Experiments with optical tweezers have shown their capability of differentiating cancerous cells from non-cancerous by using two non-focused laser beams in opposite directions (see [4]). Figure 1: Kinesin dimer attached to a microtubule. Picture taken from [5]. 3 See the original quotation in the beginning of the next section. 3

5 3 Optical Trapping "I decided to try to see radiation pressure. I made a calculation of how much it would be on a small transparent sphere. That started the whole business for me. What I did was focus a beam down on little spheres in water and watched as they were pushed along and mysteriously collected at the chamber wall. I tried to understand this and figured it out using simple ray diagrams. Then, I replaced the glass wall with another opposite beam to hold the particles in place with just light. I tried it and it worked. This was the first optical trap. It turned out to be a pretty important discovery." 3.1 Optical trapping theory Arthur Ashkin, 1997 When describing light, we frequently speak of duality principle, meaning that light can be treated as particles or as waves. In the first case, light rays can be described as a series of consecutive fundamental particles called photons, each carrying a linear momentum which is determined by the wavelength of light p = E c = h λ, (1) where E denotes the energy of each photon. The momentum is a vector quantity and points in the direction of light propagation. When reflected or refracted, the change in the direction or amplitude of the momentum imparts a continuous force on the medium the ray reflects or refracts on. A focused laser beam which is capable of trapping small particles in three dimensions is referred to as an optical trap. In a macroscopic world the radiation pressure is so small in comparison with other forces, such as gravity, that it can be safely disregarded. For example, a laser produces dn dt = P (2) hν photons per second. When reflected by a mirror, they produce a constant momentum rate of dp dt = 2dN dt p = 2dN dt which, in turn, causes the mirror to accelerate. dp dt = mdv dt h λ, (3) a = 1 dp m dt 1 P m c. (4) For a 1 W laser and a 1 kg mirror the acceleration is of order of g, where g is the gravitational acceleration. Thus, the radiation pressure can be clearly neglected when dealing with macroscopic particles. On the other hand, a 1µm cube has a mass of 1 pg, assuming a density of water. If the light is completely focused so that it reflects of this particle, it produces an acceleration of 10 5 g. 4

6 Therefore, it seems that the radiation pressure is large enough to be able to move microscopic particles. In case when the wavelength of the light used to trap particles is much smaller than the diameter of the particle (λ d), the effects of diffraction can be neglected and simple ray optics treatment can be used to calculate the forces exerted on the particle. On the other hand, if the wavelength of the light is much larger than the diameter of the particle (λ d), the trapped particle can be treated as an electric dipole interacting with the electromagnetic field of laser light. Calculations done in this limit are referred to as the Rayleigh regime. However, in real experiments the particle diameters are of order of the wavelength of the light used for trapping. Thus, none of the above limits can be justified 4. But since both limits provide a simple explanation of how an optical trap works on a quantitative level, they will be presented. 3.2 Ray optics regime If the particle diameter is much larger than the wavelength used to trap it, the direction of the resorting force can be qualitatively explained by drawing two paths of rays with different intensities refracting in the trapped spherical particle. If the particle is displaced laterally, as in Figure 2, the force caused by refraction of a ray with higher power density points in the opposite direction compared to the force of a ray with lower power density. However, since the force is linearly proportional to the power density of the ray, the resulting force points in the direction towards the trap center. If the particle is displaced in the opposite direction, the force also points towards the trap center due to the symmetry of the problem. It is essential to notice that the restoring force is caused because of the radial gradient of the power density of the laser light. Thus, this force is usually called gradient force. The refraction also exerts a force component pointing in the axial direction, as in Figure 3, again pointing towards the focus. The two figures do not take into account the force caused by partial reflection of the beam off the particle (the scattering force). This force tends to move the particle in the +z-direction out of the trap. If this force is small enough, the gradient force and the scattering force can sum up to zero resulting in an equilibrium position of the particle slightly above the beam waist. If the scattering force is too large, the particle escapes from the trap. Thus, it is important that the reflection is small, i.e. the particles must be transparent. By using Fresnel equations for coefficients of reflection R and transmission T 5, the optical forces on a sphere can be calculated (see [8]). The scattering force can be expressed as F s = n 1P c ( 1 + R cos 2θ T ) 2 cos(2θ 2φ) + R cos 2θ 1 + R 2 + 2R cos 2φ = n 1P Q s, (5) c where n 1 is the index of refraction of the suspending medium, θ and φ the angles of incidence and refraction. The angles φ and θ are related by Snell s law n 2 = sin θ n 1 sin φ, (6) 4 An article by Svoboda and Block [6] includes a review of the calculations in all-size regimes. 5 See, e.g. Hecht, E., Optics. 5

7 Figure 2: A qualitative plot of the gradient forces acting on a laterally displaced particle. The scattering force is not shown. Picture taken from [7]. Figure 3: A qualitative plot of the gradient forces acting on an axially displaced particle. The scattering force is not shown. Picture taken from [7]. 6

8 where n 2 is the refractive index of the object. Similarly, the gradient force equals ( R sin 2θ T F g = n 1P c 2 sin(2θ 2φ) + R sin 2θ 1 + R 2 + 2R cos 2φ ) = n 1P Q g. (7) c By using vectorial addition of the two forces, it can be concluded that a ray of power P exerts the force of magnitude F t = n 1P c Q 2 s + Q 2 g = n 1P c Q(θ, n 2 n 1, R, T ). (8) The total force exerted on a particle can be obtained by summing over all rays passing through it. The force ranges from 1 to 100 pn, depending on the size of the trapped particle, indices of refraction, and the power used to form a trap. 3.3 Rayleigh regime If the diameter of the particle is much smaller than the wavelength of the laser light, the trapped particle can be approximated with point dipoles, since the electromagnetic field is practically constant from the particle s point of view. The scattering force can be expressed as S σ F s = n 1, (9) c where S is the time-averaged Poynting vector of the electromagnetic radiation and σ = σ(d, λ, n1 n 2 ) the effective scattering cross-section of a particle with diameter d. The gradient force is actually the Lorentz force acting on the dipole induced by the electromagnetic field F g = α 2 E2, (10) where E is the electric field and α = α(d 3, n2 n 1 ) the polarizability of the particle. The force is highly dependent on the size of the particle and usually has a femtonewton-range value. 7

9 4 Basic Design 4.1 General description A basic laser tweezers system comprises of a low-power infrared laser and a set of mirrors and focusing lenses. Such a system is capable of manipulating cells, beads attached to various objects and other micrometer-sized particles. Its usability is limited by the laser power and vibration annullation capability. On the other hand, a more complex system additionally includes higher laser power, mechanical stability control and a real-time monitoring system. 4.2 Microscope Optical tweezers can be mounted to a high-quality video microscope with an epiillumination port. However, there are three items that need additional attention, namely a dichroic mirror, which reflects the infrared laser light and transmits the visible light, a high numerical aperture objective and a mechanically stable system. The infrared laser light and the light used for video imaging share the same path between the dichroic mirror and the trap. Therefore, the dichroic mirror must be mounted below the objective and allow the visible light used for video recording to pass through and reflect the infrared light coming from the laser. If the reflectivity of the dichroic mirror as a function of the wavelength is known, the laser beam wavelength should be at the maximum of the reflectivity function to reflect as much power as possible. Although probably not designed for infrared wavelengths, an objective with a numerical aperture of more than 1.0 usually suffices to form a stable trap. If the trap is formed too far away from the objective (more than 20µm), the focusing angle might not be large enough for the gradient force to nullify the scattering force and the particle will escape from the trap. Since almost half of the light power is absorbed by the objective, a too high laser power can damage the optical components. Objective apertures are sometimes made of black plastic which reduces image quality and is very hard to remove if illuminated by too high light power density. In order to reduce irradiation damage to lenses and other optical components manufacturers typically recommend the maximum laser power of 1 W. For accurate measurements of forces, the optical tweezers must be capable of producing controlled movements in steps of several nanometers. Although motorized mechanical stages prove useful for bringing the particles over larger distances, they have a step size of several hundred nanometers or more. Thus, they are impractical for measuring forces in a trap. On the other hand, piezoelectric positioners can impart the movements of particles in required step size range. Anyhow, if there is any relative motion between the video camera and the microscope, the distances cannot be accurately measured. 4.3 Laser The laser wavelength used to trap particles should be carefully chosen, especially if dealing with in vivo experiments. If too small wavelength, e.g. visible light, is used, the biological material highly absorbs this wavelength range and 8

10 quickly overheats. On the other hand, longer wavelengths are absorbed by the surrounding water. Therefore, the wavelength region which is usually used for optical traps lies between 780 and 1100 nm. To sum up, the trapped particle must be dielectric, transparent, and needs to have the index of refraction larger than the surrounding medium. However, when doing in vivo experiments, we do not have much choice regarding the material. In this case, some adjustments of wavelength, power, and focusing might be needed. In order to form a stable optical trap a low noise gaussian beam laser with continuous output must be used. Gaussian beam lasers are commonly referred to as TEM 00 lasers. The light coming from such laser can be focused to an area with the diameter of order of the wavelength to form a diffraction-limited spot. Another possibility is to use a Bessel beam which can be produced by passing a Gaussian beam through a conical lens. Bessel beams are better in comparison to Gaussian ones because the central peak has a smaller width which in turn reduces diffraction effects thus yielding better results. 4.4 Classification by number of beams Single beam optical tweezers use a highly focused laser light in order to generate sufficiently high trapping forces and overcome the scattering force which is caused by the reflection. On the other hand, two laser beams in opposite directions can generate much larger forces since the two scattering forces approximately cancel out. Another advantage is that the light does not need to be extremely highly focused. The disadvantage of a dual beam system is its complexity. Usually, such a system cannot be integrated into a classical video microscope, so the essential parts required for monitoring must be constructed separately. Additional problems occur with alignment of the beams. The foci of the two beams must coincide up to one micrometer accurately, which usually presents quite a challenge to the constructor. Thus, single beam systems are more often used for submicrometer-sized particles. 9

11 Figure 4: A simplified scheme of optical tweezers. Picture taken from [9]. 5 Force Measurement Calibration 5.1 Viscous drag force calibration Since neither the ray optics treatment nor the Rayleigh approximation provided satisfactory results, the optical trap calibration must be performed experimentally. Calibrating an optical trap means determining the force/displacement relation. A viscous drag force can be obtained by producing a steady and essentially laminar flow of surrounding medium, which exerts a constant force on the trapped particle. For a spherical particle, this force can be calculated by using Stokes law F = 3πηdv = γv. (11) The drag coefficient is denoted by γ, η is the viscosity of the surrounding fluid, d the trapped particle diameter and v the speed of the medium. 5.2 Brownian motion calibration Instead of actively producing a force on the trapped particle, the optical trap potential can be determined by just monitoring particle s position over time, which changes due to the Brownian motion. If the particle is not trapped, it exhibits a random motion, so that the average distance x increases with sqare root of time t. Var(x) = x 2 (t) x(t) 2 = 2k BT γ t = 2Dt. (12) D denotes the diffusion constant, k B Boltzmann s constant and T the temperature. 10

12 When trapped, the particle experiences diffusion forces and forces of the optical trap, which confine the particle to a limited volume. If we assume that the trapping force linearly depends on the displacement of the particle, the net time-dependent force can be written as (Langevin equation) F (t) = γ dx + κx. (13) dt The time-average of F (t) is zero since the particle is trapped in a stable equilibrium. Its power spectrum, S F (f), contains in case of idealized Brownian motion only white noise, i.e. all frequencies have equal amplitudes. S F (f) = F 2 (f) = 4γk B T, (14) F(f) is the Fourier transform of F (t). The Fourier transform of Langevin equation gives a Lorentz-like spectrum with characteristic frequency f c = κ/2πγ. S x (f) = ξ 2 (f) = k B T γπ 2 (f 2 c + f 2 ). (15) The frequencies, which are lower than f c have approximately equal amplitudes, whereas at the high-frequency limit the amplitude falls with the square of the frequency. The high-frequency behavior can also be observed in free diffusion, which implicates that at short time scales diffusion effects prevail over trapping forces. Figure 5: Frequency spectrum of the trapped particle. Picture taken from [10]. 5.3 Conversion to physical units After experimental determination of parameters S 0 and f c the trap stiffness κ can be calculated as κ = 2k BT πs 0 f c or κ = 2πγf c. (16) The trap stiffness has the same units as the spring constant in Hooke s law. The linear relation between the force and the displacement is usually a good approximation for less than 100 nm displacements. 11

13 6 Applications Previous sections of the seminar have shown that optical tweezers are capable of exerting forces in range of piconewtons on micrometer- and submicrometer-sized dielectric particles and measure distances accurately down to nanometer range. The true power of such an instrument can be demonstrated by performing in vivo experiments. Since it became possible to attach a single molecule to a trapped bead, the forces acting on the molecule under various conditions can be accurately measured. There are two major advantages of such an experimental design, namely the isolation of a single molecule vastly simplifies the conditions in which the experiment is taking place. The interactions which would have taken place otherwise would only disturb the experiment. Since the properties of a single molecule can be directly measured, there is no need for configurational averaging anymore. Furthermore, the obtained results can be used to model physical behavior of a single molecule. Although optical tweezers can be used for doing various experiments, the next section focuses primarily on their usage in applicative microbiological research. Additional knowledge of the range of forces used in DNA transcription of Escherichia Coli bacteria helps us understand DNA replication process more thoroughly. Attached to polystyrene beads, a single molecule can be studied without the disturbances from the environment. By stretching an RNA molecule, its structure and thermodynamics of folding can be deduced (see [11]). Measurements of velocity/force DNA transcription profile have uncovered new information about the dynamic structures and biophysical properties of DNA (see [12]). Similar experiments gave a better insight into the kinetics of DNA polymerase (see [13]). In another study, the packaging of DNA into a virus capsid has been directly observed. Figure 6: The structure of part of a DNA double helix. Picture taken from [14]. 12

14 6.1 Studying molecular motors Studies of DNA replication of Escherichia Coli bacteria have received quite much interest among experimental scientists working with optical tweezers. The range of forces an optical trap is capable of producing has proven useful for measuring the treshold tension that can be exerted on a single DNA molecule to stop the DNA copying process. The enzyme, which produces a force on the DNA required to start the copying process is called RNA polymerase. Since it uses energy to move the DNA it belongs to a group of microbiological motors. The process of copying takes place in two phases. First, the RNA polymerase copies the DNA sequence to create a single-stranded messenger RNA in a process called transcription. Then, a ribosome uses this messenger RNA to produce a specific protein in a process referred to as translation. The velocity/force profile plotted on a graph in Figure 8 shows a velocity plateau up to a specific treshold force of about 25 pn, where the transcription velocity reduces rapidly and can even entirely stop if too large forces are exerted on the DNA. Anyhow, the process can continue if the force is reduced. The figure shows that even under a constant force below the treshold value the motor occasionally stops and resumes the transcription process after a while. The pauses are usually less than a minute long. The transcription velocity as a function of the force acting in the opposite direction of transcription motion can be measured by attaching a single DNA molecule onto a spherical bead which is held in position by using an optical trap. The RNA polymerase enzyme is attached to another bead which is fixed onto a micropipette. The force is exerted by moving the micropipette in the appropriate direction. Figure 7: Left: low-temperature electron micrograph of a cluster of Escherichia Coli, magnified 10, 000 times. Each individual bacterium is oblong shaped. Right: schematic diagram of an optical tweezers experiment to measure the transcription forces generated by Escherichia Coli RNA polymerase. Both pictures taken from [10]. 13

15 Figure 8: Left: time dependence of number of transcribed base pairs of Escheriachia Coli DNA molecule below the treshold force. Right: speed of transcription as a function of the force acting on a DNA in the opposite direction. Force and speed scales are normalized based on the treshold force and half of the unloaded speed, respectively. Both pictures taken from [10]. 6.2 Manipulating small objects with holographic traps The original discovery by Ashkin in 1970 provided the fundamentals of optical trapping. A constantly increasing interest in usage of optical tweezers in various experiments resulted in a growing need for more complex multiple-trap systems which are capable of manipulating particles of various sizes simultaneously. This can be accomplished by using holographic optical tweezers. The device itself is quite similar to the basic optical tweezers with a single laser beam. The main difference is that instead of sending the laser light directly through the objective, holographic optical tweezers pass the light through a spatial light modulator first. This is a liquid crystal display, which modulates the phase of the laser light differently at different locations. Thus, it forms a hologram, which is then passed through the objective of the microscope to form multiple optical traps at desired locations in three dimensions. Consequently, the power per trap is reduced. Figure 9: Left: an array of µm silica spheres. Based on the capability of trapping such a large number of particles one can depicts their use in microconstruction. Right: Radially moving array of traps sweeps particles from the center of the screen. By re-running the process several times a clean working area is created but the particles remain close enough for easy access. The particles are the same as the ones on the left picture. Both pictures taken from [15]. 14

16 Figure 10: Left: Different species of particles can be sorted by using an array of optical traps which are continually moving down to push larger particles into the channel D and allowing smaller particles to flow in the channel B. Right: Bonding and separating yeast cells demonstrates the potential for affinity studies. Both pictures taken from [15]. 7 Conclusion This seminar presents a basic qualitative and quantitative explanation of how an optical trap works and lists the most essential parts needed to construct a basic optical tweezers. It also shows that transition from the discovery of optical trapping to multiple-trap holographic optical tweezers is not a trivial extension but a years-long process. Trapping force magnitude calculations and calibration methods show the usefulness of optical tweezers for doing various tasks from manipulating spherical beads of micrometer-ranged radius to exerting extremely small forces on molecules. Due to the flexible force range which extends over several orders of magnitude one can easily envision the use of optical tweezers ranging from sorting small particles to performing a controlled chemical reaction with picoliters of reagents. Thus, it seems that optical tweezers have a bright future. 15

17 8 Acknowledgments I would like to thank my advisor doc. dr. Igor Poberaj for a thorough and critical revision of this seminar from all aspects and for kindly providing the literature which is simple enough to cover the very basics of optical tweezing and general at the same time so that the reader can envision the variety of practical applications where optical trapping comes handy. 16

18 References [1] A. Ashkin. Acceleration and trapping of particles by radiation pressure. Physical Review Letters, 24: , [2] A. et al Ashkin. Optical trapping and manipulation of viruses and bacteria. Science, 235:1517, [3] A. Constable. Demonstration of a fiber-optical light-force trap. Optics Letters, 18:1867, [4] V. Jamieson. Bending light backwards. New Scientist, 2386, [5] March 17, [6] K. Svoboda and S. M. Block. Biological applications of optical forces. Annual Review of Biophysics and Biomolecular Structure, 23: , [7] J. Mameren. Single molecule mechanics of biopolymers: an optical tweezers study joost/thesis.pdf. [8] A. Ashkin. Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime. Laser Tweezers in Cell Biology, [9] March 17, [10] M. Williams. Optical tweezers: Measuring piconewton forces [11] C. Bustamante. Single-molecule studies of dna mechanics. Current Opinion in Structural Biology, [12] J. F. et al Allemand. Stretched and overwound dna forms a pauling-like structure with exposed bases. The Proceeding of the National Academy of Science, [13] B. Maier. Replication by a single dna polymerase of a stretched singlestranded dna. The Proceeding of the National Academy of Science, [14] March 17, [15] J. Plewa. Prospects for holographic optical tweezers