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A Practical Guide to Optical Trapping Joshua W. Shaevitz jshaevitz@berkeley.edu

A Practical Guide to Optical Trapping Joshua W. Shaevitz jshaevitz@berkeley.edu October 5, 2009 2 Chapter 1 Introduction to optical trapping In the last few decades, novel microscopy techniques have been developed to monitor the activity of single enzymes as they perform their biological functions in vitro. Motor proteins such as kinesin, myosin, F1Fo ATPase, and RNA polymerase have been mercilessly subjected to magnetic, elastic, and optical forces [14, 40, 48, 16, 18]. In 1986, Ashkin and colleagues reported the ﬁrst observation of a stable three-dimensional optical trap, or optical tweezers, created using radiation pressure from a single laser beam [4]. Only a few years later, Block and colleagues had used an optical trap to manipulate and apply forces to E. coli ﬂagella [8] and single kinesin motors [9]. Optical traps use light to manipulate microscopic objects as small as 10 nm using the radiation pressure from a focused laser beam. In addition, measurement of the light deﬂection yields information about the position of the object in the laser focus. Many excellent reviews have been written about optical trapping, its uses, and designs, see e.g. [2, 6, 22, 27, 37, 38, 43]. In particular, Lang and Block [23] is a thorough review of the optical trapping literature. This manuscript is meant to be a practical guide to understanding optical traps, and not an in depth review. When possible, simple examples and explanations are used to give the reader an intuitive feel for how these systems work and how they are implemented. I hope that this document will continue to improve, and welcome any comments. The picoNewton and nanometer ranges of force and distance accessible to optical traps make them particularly useful for studying biological systems (Fig. 1.1) [7]. Optical forces have been used to investigate structural properties of biological polymers such as DNA [10, 46], membranes [39], whole cells [3] and microtubules [21]. Microrheological properties of these objects can be probed through the application of forces either to the object itself, or to a small dielectric sphere, or bead, to which the object is attached. Molecular motors represent the most used application of optical traps in the biological sciences. A great deal has been learned about kinesin [1, 9, 11, 12, 19, 20, 44], dynein 3 4 CHAPTER 1. INTRODUCTION TO OPTICAL TRAPPING [26, 25], myosin [28, 32, 33, 41, 42], and RNA polymerase [13, 15, 30, 36, 45] using optical forces. 5 Figure 1.1: Diﬀerent Opti- cal Trapping Assays. (A) Optical trapping studies of RNA polymerase typically ﬁx the polymerase to a op- tically trapped bead while the distal end of the DNA is attached to the micro- scope coverslip. As the polymerase moves along the DNA it must do work against the optical trap. (B) In a typical assay of kinesin motion, a motor– coated bead moves along a microtubule towards its plus–end, while being sub- jected to a retarding force by the optical trap. (C) Studies of non-processive motors, such as muscle myosin and NCD, often in- volve more complicated ge- ometries and multiple op- tical traps. In this study, a myosin motor ﬂeetingly grabs the actin ﬁlament suspended between the two optical trapped beads and strokes before letting go. (D) Optical traps can also be used to study the poly- merization of bioﬁlaments. Here a polymerizing micro- tubule is immobilized by attachment to two opti- cally trapped beads. As the microtubule grows it rams against a glass pillar pushing against the optical forces. 6 CHAPTER 1. INTRODUCTION TO OPTICAL TRAPPING Chapter 2 How optical traps work In the focus of a laser beam a dielectric particle, such as a glass or polystyrene bead, experiences a force, called the gradient force, that tends to bring push towards the laser focus where the light intensity is highest. This force arises from the momentum imparted to the bead as it scatters the laser light. Al- though the full theory of optical trapping is quite complex (see e.g. Rohrbach and Stelzer [34]), a few simpliﬁed examples allow for a good working intuition. The easiest case to consider occurs when the particle is much larger than the wavelength of light and is displaced from the laser focus laterally (Fig. 2.1A). When the particle sits to the right of the laser focus, f, the overall direction of propagation of the laser beam is deﬂected to the right. Rays a and b are refracted such that they meet to the right of the laser focus. The momentum change of these photons imparts an equal and opposite momentum change to the particle. The force on the particle at a particular displacement from the focus is linearly proportional to the total laser power – the more rays that are diﬀracted, the more force is imparted to the particle. The situation gets slightly more complicated when one considers the more realistic case of a Gaussian laser mode, one in which the intensity proﬁle of the laser beam in a plane perpendicular to the direction of propagation is a two–dimensional Gaussian (Fig. 2.1B). When the dielectric particle is very small compared to the wavelength of light it can be approximated as a perfect dipole that feels a Lorentz force due to the gradient in the electric ﬁeld (Fig. 2.1C). Because the beam proﬁle is Gaussian, the Lorentz force points towards the focus and is equal to F = (p · ▽) E + 1 c dp dt × B (2.1) where p = αE is the dipole ﬁeld and α is the polarizability. Optical traps are typically used with a continuous wave (CW) laser such that ∂ ∂t (E × B) = 0 . 7 8 CHAPTER 2. HOW OPTICAL TRAPS WORK Figure 2.1: Simpliﬁed illustrations of optical trapping. (A) The simplest ray- optics diagram. In the absence of the bead, two rays (a and b) are focused through the objective lens to position f, the true laser focus. Refraction through the bead, which is displaced to the right of the laser focus, causes the new focus to lie to the right of f. After exiting the bead, ray a is bent up and to the right of its original trajectory, while ray b is deﬂected down and to the right. Fa and Fb represent the forces imparted to the bead by rays a and b; Ftotal is the sum of these two vectors and points to the left. (B) The force from a single-beam gradient optical trap with Gaussian intensity proﬁle; two rays are drawn. The central ray, a, is of higher intensity than the extreme ray, b. Again, the bead is displaced to the right of the true laser focus. The total force on the bead, Ftotal, again points to the left. (C) Dielectric particles much smaller than the wavelength of light can be considered to be perfect dipoles. The gradient in intensity, and hence electric ﬁeld, produces a Lorentz force on the particle directed towards the laser focus. 9 In this case the time-averaged force becomes ⟨F ⟩= α 2 ▽⟨E2⟩ (2.2) Typically, optical trapping experiments are performed using 500–nm polystyrene or glass spheres and a 1064–nm wavelength trapping laser. This combination of bead size and laser wavelength puts the real physics somewhere between the ray–optic and dipole regimes. The full theory of Mie scattering can get quite complicated, but the intuition gained from the other representations shown in Figure 2.1 remains useful even if it is not completely accurate. A restoring force also exists in the axial dimension. If the particle is dis- placed axially below the laser focus (Fig. 2.2A), the overall direction of the laser propagation is not changed, but the divergence is. Rays a and b are refracted such that the new focus within the bead lies below f, and are more convergent upon exiting the bead. This slight refocusing of the laser causes a force on the particle pointing upwards, towards the laser focus f. The op- posite is true when the particle is above the focus and the rays become more divergent (Fig. 2.2B). Not all of the light is refracted through the particle; some gets reﬂected backwards. The force associated with these rays, the scattering force, pushes the particle away from the laser focus and causes the center of the optical trap to exist at a position displaced axially from the focus. 10 CHAPTER 2. HOW OPTICAL TRAPS WORK Figure 2.2: Description of axial trapping forces. Axial displacements of a bead in an optical trap change the relative amount of divergence of the focused laser light. In the absence of the bead, two rays (a and b) are focused through the objective lens to position f, the true laser focus. (A) Refraction through the bead, which is displaced below the laser focus, causes the new focus to lie below f. Upon exiting the bead the two rays are more convergent; ray a is bent down and to the left, while ray b is deﬂected down and to the right. Fa and Fb represent the forces imparted to the uploads/Geographie/ ot-practicle-guide.pdf