MT Striped Birefringence Pattern Formation and Application of Laser Tweezers in Microrheology, Bacterial Motility and Adhesion by Yifeng Liu M. Sc., Brown University, 2005 B. Sc., University of Science and Technology of China., 2003 Thesis Submitted in partial fulllment of the requirements for the degree of Doctor of Philosophy in the Department of Physics at Brown University Providence, Rhode Island May, 2009 Abstract of MT Striped Birefringence Pattern Formation and Application of Laser Tweezers in Microrheology, Bacterial Motility and Adhesion by Yifeng Liu, Ph.D., Brown University, May, 2009. Various mechanisms govern pattern formation in chemical and biological reaction systems, giving rise to structures with distinct morphologies and physical properties. The self-organization of polymerizing microtubules (MTs) is of particular interest be- cause of its implications for biological function. We report a study of the microscopic structure and properties of the striped patterns that spontaneously form in polymer- izing tubulin solutions and propose a mechanism driving this assembly. We further present a mechanical model to explain the process and mechanism of buckling, and to infer properties of MT bundles such as packing geometry and size. Using oscillatory optical tweezers based active microrheology, the frequency-dependent viscoelasticity of MT bundled networks were measured. This method implements forced oscillation of a 1.5µm silica bead embedded in a MT bundled network. Both the storage modulus and the loss modulus depend on the direction of the oscillatory motion of the bead relative to the alignment direction of the bundles. Furthermore, we have built an infrared laser tweezers setup for passive microrheology and to study the biophysics of bacterial motility and adhesion. We determined the force prole both near the center of the trap and at the far edge. Using the newly-built laser tweezers and position detection system, we studied microrheological properties of F- actin in both isotropic and nematic phases. This method records the displacement of thermally driven micron-sized beads, based on which shear moduli of the underlying network were calculated. The result is consistent with that obtained by video particle tracking. In addition, we use the calibrated optical trap to determine the trapping force on a swimming Caulobacter crescentus swarmer cell during its escape from the trap cen- ter. We further apply laser tweezers to study the rotation of a trapped Caulobacter swarmer cell and the process by which the cell struggles to escape from the trap center. Aided by the laser trap, we also show that bringing the holdfast of a stalked cell close to a glass surface facilitates adhesion of the cell to the surface, consistent with the model that cells must overcome a repulsive barrier at the surface to adhere eciently. © Copyright 2009 by Yifeng Liu Vita Education ˆ Brown University, Providence, RI 02912 2003-2008  Ph.D. in Physics. Department of Physics. September 2008  M.Sc. in Physics. Department of Physics. May 2005 ˆ University of Science and Technology of China, Hefei, P.R. China 1999-2003  B.Sc. & Eng. in Materials Science and Engineering. Department of Materials Science and Engineering. July 2003 Publications and Manuscripts iv ˆ Y. Liu, Y. Guo, J. M. Valles Jr. and J. X. Tang, Microtubule bundling and nested buckling drive stripe formation in polymerizing tubulin solutions Proc. Natl. Acad. Sci. U.S.A., 103, 10654 (2006) ˆ Y. Guo, Y. Liu, J. X. Tang and J. M. Valles Jr., Polymerization Force Driven Buckling of Microtubule Bundles Determines the Wavelength of Pat- terns Formed in Tubulin Solutions PRL, 98, 198103 (2007) ˆ Y. Guo, Y. Liu, R. Oldenbourg, J. X. Tang and J. M. Valles Jr., Eects of Osmotic Force and Torque on Microtubule Bundling and Pattern Formation submitted ) ( ˆ Jun He, Michael Mak, Yifeng Liu and Jay X. Tang, Counter-ion Dependent Microrheological Properties of F-actin Solution across Isotropic-Nematic Phase Transition Phys. Rev. E , 78, 011908 (2008) ˆ Yifeng Liu, Guanglai Li, Anubhav Tripathi, Yves V. Brun, and Jay X. Tang, Application of laser tweezers in the studies of bacterial swimming, cell body rotation, escape and adhesion ( submitted ) Experiences ˆ Research Assistant. Advisor: Professor Jay X. Tang, Department of Physics, Brown University, Providence, RI. May 2004 - August 2008. ˆ Teaching Assistant. Department of Physics, Brown University, Providence, RI. September 2003 - May 2004. Preface This thesis presents two biophysics studies, one is on microscopic investigation and mechanical modeling of Microtubule (MT) striped birefringence pattern formation in vitro, the other is about the construction of an optical tweezers and position detection system and its applications in microrheology and bacterial motility and adhesion. These two studies are presented as two separated parts. In part one, I rst give an introduction to MT and MT birefringence patterns in Chapter 1. Chapter 2 describes the experimental methods and techniques used in this study including sample preparation, various microscopic methods and details about certain measurements. In Chapter 3, the macroscopic and microscopic struc- ture, and physical properties of MT striated birefringence patterns are presented. A nested buckling model is also proposed to explain the undulations of birefringence and uorescence intensity. These results were published in [57]. Based on the ob- served pattern structures and further studies on the growth of bundles and solution retardance, we propose a mechanical model for the buckling process in Chapter 4 and the results were published in [42]. The results regarding the eects of PEG on MT vii bundling and pattern formation has been submitted for publication [41]. In part two, I rst gave an introduction to optical tweezers and microrheology in chapter 4. In chapter 5, I described the construction of the optical tweezers and position detection system, and the process of calibrating the optical trap. In chapter 6, the application of optical tweezers in microrheology is presented, including exper- imental methods and results. In chapter 7, the application of optical tweezers on bacterial motility and adhesion is presented. Yifeng Liu Acknowledgments The rst part of the work presented on MT pattern formation in this thesis is a collaborative project with Yongxing Guo under the joint supervision of Prof. Jay X. Tang and Prof. James M. Valles Jr. I would like to express my great thanks to my thesis advisor, Prof. Jay X. Tang, for his supervision and insightful advice throughout this work. My great thanks also go to Prof. James M. Valles Jr. for his kind support and valuable discussions. I would also like to thank Yongxing Guo for collaborating in the MT pattern formation project. I would like to recognize all former and current lab members including: Dr. Karim Addas, Dr. Guanglai Li, Dr. Jorge Viamontes, Dr. Jing Wang at Lehigh University, Dr. Qi Wen, Jun He, Hyeran Kang and Patrick Oakes. Special appreciation also goes to Prof. Allan F. Bower, Prof. Yves V. Brun at Indiana University, Prof. Rudolf Oldenbourg at Marine Biology Lab, Prof. Daniel OuYang at Lehigh University, Prof. Thomas Powers, Prof. Anubhav Tripathi for their helpful suggestions. I also want to thank my parents and friends for their encouragements and support throughout my experience at Brown University. ix Contents Vita iv Preface vii Acknowledgments ix List of Figures xv I Investigation of Mechanism and Characteristics of Micro- tubule Striated Birefringence Patterns 1 1 Introduction to Microtubule (MT) and MT Pattern Formation 2 1.1 What is microtubule . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Motivation and importance of studies on MT assembly and pattern formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Experimental Methods 8 2.1 Experimental chamber . . . . . . . . . . . . . . . . . . . . . . . . . . 8 x 2.2 Purication of tubulin and polymerization of MTs . . . . . . . . . . . 8 2.3 Kinetics measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Aligning MTs with static magnetic elds . . . . . . . . . . . . . . . . 10 2.4.1 Magnetic torque on a single MT . . . . . . . . . . . . . . . . . 11 2.4.2 Time needed to align a MT in a magnetic eld . . . . . . . . . 11 2.5 Fluorescence and phase-contrast microscopy . . . . . . . . . . . . . . 12 2.6 Birefringence measurements . . . . . . . . . . . . . . . . . . . . . . . 13 2.7 MT bundle size measurement using PolScope . . . . . . . . . . . . . . 14 2.8 Sedimentation and quantitative uorescence assay . . . . . . . . . . . 14 3 Properties of Microtubule Striped Birefringence Patterns 16 3.1 Macroscopic appearance of birefringence patterns . . . . . . . . . . . 16 3.1.1 Role of convective ow and gravity in pattern formation . . . 19 3.2 MT bundle formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Small angle x-ray scattering experiments and data analysis . . . . . . 26 3.4 Birefringence measurement, phase contrast and uorescence imaging of striated patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4.1 Quantitative analysis of periodic variations in birefringence and uorescene intensity . . . . . . . . . . . . . . . . . . . . . . . 30 3.5 Decomposition of the pattern structure into MT bundles and dispersed MTs in the network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 xi 3.6 Estimation of MT bundle size and bending energy based on retardance measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.7 Time evolution of retardance, bundle length and buckling wavelength 38 4 Mechanism of Striped Pattern Formation and a Mechanical Buckling Model 43 4.1 Mechanism of striped birefringence pattern formation . . . . . . . . . 43 4.2 Estimation of buckling wavelength based on minimization of energy . 44 4.3 The mechanical buckling model . . . . . . . . . . . . . . . . . . . . . 46 4.3.1 Buckling force . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.4 Numerical estimates about buckling wavelength . . . . . . . . . . . . 49 II Optical Tweezers and Its Applications in Microrheology and Caulobacter Motility and Adhesion 53 5 Introduction to Optical Tweezers and Microrheology 54 5.1 How optical tweezers work . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Microrheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2.1 Passive microrheology . . . . . . . . . . . . . . . . . . . . . . 57 5.2.2 Active microrheology . . . . . . . . . . . . . . . . . . . . . . . 60 6 Setup of the Optical Trapping and Position Detection System 63 6.1 Optical tweezers setup . . . . . . . . . . . . . . . . . . . . . . . . . . 63 xii 6.2 Position detection using photodiode detector . . . . . . . . . . . . . . 65 6.3 Calibration of the optical trap . . . . . . . . . . . . . . . . . . . . . . 66 6.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.3.2 Experimental determination of trap stiness and whole range trap force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7 Application of Optical Tweezers in Microrheology of Microtubule and Actin Networks 75 7.1 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.1.1 Measurement of viscoelasticity of isotropic and nematic actin networks using passive microrheology . . . . . . . . . . . . . . 75 7.1.2 Measurement of anisotropic elastic properties of MT bundled network using active microrheology . . . . . . . . . . . . . . . 79 7.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.2.1 Moduli as a function of frequency for isotropic and nematic F-actin solutions . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.2.2 Anisotropic elastic properties of bundled MT network . . . . . 88 8 Application of Optical Tweezers in the Study of Bacteria Swimming and Adhesion 91 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.2 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . 93 xiii 8.2.1 Bacteria sample preparation . . . . . . . . . . . . . . . . . . . 93 8.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 94 8.3.1 Trapping force on Caulobacter swarmer cells . . . . . . . . . 94 8.3.2 Cell body rotation of a trapped Caulobacter swarmer cell . . . 95 8.3.3 Escape of a Caulobacter swarmer cell from the laser trap . . . 98 8.3.4 Facilitated adhesion of a Caulobacter stalked cell via holdfast using laser tweezers . . . . . . . . . . . . . . . . . . . . . . . . 102 8.4 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 105 Bibliography 109 xiv List of Figures 1.1 Structure of a microtubule (Adapted from http://www.med.unibs.it). 3 1.2 Schematic drawing of a mitotic spindle (adapted from www.sparknotes.com). Microtubules aligns and segregates chromosomes during eukaryotic cell division. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 GTP-Cap Model for Microtubule Dynamic Instability. The polymer- ization phase is thought to be stabilized by a thin cap of tubulin dimers at the microtubule plus-end with a GTP molecule associated with each beta-tubulin subunit. Stochastic loss of the GTP cap, due to hydroly- sis or subunit loss, results in a transition to the depolymerizing phase known as catastrophe. Back transition (rescue) is also observed. Taken from Inoue and Salmon (1995) Mole. Biol. Cell 6:1619-1640. . . . . . 4 3.1 MT striped birefringence pattern observed between crossed polarizers. The axis of the cuvette was at 0o (left) and 45o (right) relative to that of the polarization directions of the polarizers, respectively. The dimensions of the cuvette are 40 x 10 x 1 mm. . . . . . . . . . . . . . 17 xv 3.2 Pattern development of a polymerizing 5 mg/ml MT solution follow- ing initial alignment by a 9 Tesla vertical static magnetic eld for 5 min. The images were taken between crossed polarizers with the long ◦ axis of the cuvette (8 mm wide, 0.4 mm thick) at 45 with respect to the polarizing directions of the polarizers. (A) Image after 5 min of polymerization in the magnetic eld. Notice that the solution ap- pears brighter than the background region outside the meniscus, which ◦ has no birefringence. Rotating the cuvette by 45 changes the image to dark (data not shown), indicating that MTs are aligned in the magnetic eld direction. (B) The same sample imaged 30 min after initiation of polymerization, showing formation of stripes which are aligned pre- dominantly in the direction perpendicular to the direction of initial bulk alignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Time evolution of a MT pattern. The images were taken between crossed polarizers with the long axis of the cuvette (8 mm wide, 0.4 mm ◦ thick) at 45 with respect to the polarizing directions of the polarizers. The arrow indicates the direction of gravity. . . . . . . . . . . . . . . 18 3.4 Time series of a MT pattern formation during the initial stage. Dye particles were added to the tubulin solution before polymerization. . 19 3.5 Wave-like patterns formed in capillaries where shear ow was purposely induced. The capillary is 500 µm wide. . . . . . . . . . . . . . . . . . 20 xvi 3.6 Successive frames of a typical phase-contrast movie showing buckling of MT bundles. The contrast of the images here was enhanced for better visualization. 8 mg/ml tubulin solution in a 40 × 10 × 1 mm cuvette was prepared as described in Materials and Methods. The sample was subjected to convective ow (induced by asymmetrical thermal con- tacts between the two edges of the cuvette with a water bath-warmed aluminum holder) for the rst 9 min and then the cuvette was laid at ◦ on the microscope stage at 30 C. . . . . . . . . . . . . . . . . . . . . 21 3.7 MT bundles detected by both birefringence and uorescence measure- ments. MTs polymerized from a 5 mg/ml tubulin solution in glass cuvettes were taken out and then applied on a coverslip and covered with a glass slide. (A) Results of the PolScope measurement. Bright- ness indicates the magnitude of retardance and pins represent the slow axis directions which are consistent with the orientations of MT bun- dles. (B) Fluorescence image of the rectangular region as indicated in A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.8 MT bundles observed using high magnication uorescence microscopy. MTs polymerized from a 5 mg/ml tubulin solution in glass cuvettes were taken out and then applied on a coverslip and covered with a glass slide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 xvii 3.9 A typical confocal image of MT bundles (green). MTs were poly- merized in glass capillaries and labeled with taxol conjugated Oregon Green. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.10 Scattering pattern from a SAXS scan. The scattering pattern contains the information on the structure of the sample. The solvent scattering has been subtracted. Dierent rings around the axis of the initial incident beam represent dierent scattering angles θ. The scattering intensity I(q) at each scattering angle was averaged over the region between the two red lines. The brightest region in the center is where the strong main beam (containing unscattered beam) hit. . . . . . . . 28 3.11 Plot of logarithm of scattering intensity log(I) versus scattering vec- tor q. The peak positions are at 0.0046, 0.028, 0.057, 0.084 inverse angstrom, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.12 Scattering from oriented solution of hydrated MTs [16]. The value of q for the rst peak is at 0.027 inverse angstrom, the second peak is at 0.055 inverse angstrom, the third peak is at 0.084 inverse angstrom. All these three peaks correspond to Fourier transform of the MT cylinder. 29 xviii 3.13 Theoretical interparticle interference function [62]. d and f corresponds to arrays of nite size with perfect hexagonal symmetry correspond- ing to regular bundles of 19 microtubules; e and g corresponds to - nite paracrystals in two dimensions corresponding to bundles of mi- crotubules (about 20 MTs per bundle) with deviations from hexagonal arrangement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 xix 3.14 Quantitative analysis of the striped patterns based on birefringence measurements [57]. (A) PolScope measurement of a typical sample self- organized in 3.3 mg/ml tubulin solution (taxol-Oregon Green added for uorescence imaging performed on the same sample). The solution in a 0.4 mm thick cuvette was initially exposed to a 9 Tesla magnetic eld for 30 min. The measurement was done after the pattern was sta- ble. The brightness indicates the magnitude of retardance and the pins show slow axis orientations. (B) Retardance values along the white line p in A (blue curve) and t (red curve) using ∆(x) = ∆0 1 + tan2 ϕ(x). The x axis starts from the left end of the white line in A. The slow axis orientations ϕ are measured values as shown in C. The only t- ting parameter is ∆0 . (C) Slow axis orientations along the same line. (D) Schematic showing that the packing density is proportional to p 1 + tan2 ϕ for nested MT bundles (each curve represents a bundle). (E) Retardance along the white line in A and t proportionally to p 1 + tan2 ϕ, where ϕ is the slow axis direction at each corresponding position. This gure is taken from [57]. . . . . . . . . . . . . . . . . . 31 xx 3.15 Quantitative analysis of the striped patterns based on uorescence measurement [57]. (A) Phase contrast image (contrast enhanced) of the same region as in Fig. 3.14 but with a smaller eld of view. Blue points are manually picked up along a MT bundle. The red curve is a t to the shape of the bundle using the sum of three sinusoidal func- tions. (B) Fluorescence image of the same region, the red curve is the same as in A. (C) The blue curve is a plot of the uorescence intensity along the red curve in B as a function of the horizontal axis, x. The red p curve is a t using I(x) = Ib + I0 1 + tan2 ϕ(x), witch yields Ib = 299 and I0 = 143. Here tanϕ(x) corresponds to the calculated slope of the contour of the bundle based on the t to its shape. (D) Linear p t of uorescence intensity to 1 + tan2 ϕ using data from C. The non-zero intercept of the line with the ordinate implies a background uorescence caused by dispersed MTs. This gure is taken from [57]. 32 xxi 3.16 Illustration and measurements of the uniform elongation of MT bun- dles [1]. (a),(b) Phase contrast images of a sample region, showing progression of the pattern over 1 h. MT bundles are discerned by the thin striations. Segments 1 through 3 are adjacent pieces of a contour followed by bundles. The segment ends are dened by ducial marks. (c) Magnied view of the region denoted by the white box in (b), show- ing an encircled ducial mark. (d) Fast Fourier Transform (FFT) of (c). (e) The radially averaged FFT intensity plotted versus the az- imuthal angle θ and t using a Gaussian function. The local bundle orientation is orthogonal to the angle at which the Gaussian t peaks. (f )-(h) Length of segments 1 (f ), 2 (g), 3 (h) as a function of time. (i) Lengths of the three segments as a function of time, normalized to their lengths at 46 min. This gure is taken from [42]. . . . . . . . . . 39 xxii 3.17 Time evolution of a MT pattern obtained by measuring the retar- dance and slow axis of the sample using a PolScope imaging system [2]. (a),(b) Retardance images of a sample region at 12 and 100 min of self-organization, respectively. The color bar shows the retardance magnitude scale and the green pins provide the slow axis orientation. The straight white lines represent the slow axis line scan position. (c) Slow axis line scan (black) and the tted slow axis orientation 2π 2π ϕ(x) = arctan{A cos[ (x + x0 )]} (red) at 100 min. (d) The dom- λ λ inant buckling wavelength λ, obtained from the tted shapes of the bundle at individual time points. (e) The length evolution of the tted bundle contour. L0 = 1544 µm is the initial unbuckled length of the bundle. The segment before the arrow designates the latent period prior to the onset of the buckling. (f ) The magnitude of the retar- dance averaged over the line as shown in (a) versus the normalized length L/L0 . This gure is taken from [42]. . . . . . . . . . . . . . . . 40 4.1 Schematic drawing to illustrate the elastic buckling model. . . . . . . 46 xxiii 4.2 A schematic showing an array of bundles embedded in an elastic medium. The blue circles represent the cross sections of representative bundles with a spacing of d. The diameter of each bundle is 2R. F indicates the force exerted on the bundles by the surrounding elastic network, which is assumed to be linear with the extent of buckling ξ as: F = αξ dx. Here α is the elastic constant of the surrounding medium and dx rep- F resents unit bundle length. The stress σ is: σ = 2R dx , and strain ξ γ can be written as: γ = d . Therefore, shear modulus G0 can be αξ dx d calculated as G0 = Fd ξ2R dx = ξ2R dx = d 2R α. So α is related to G0 as √ α= 2R d G0 = η G0 , where η is the volume fraction of bundles. . . . . 50 5.1 Schematic drawing to illustrate origin of optical force in the regime where ray optics can be used. The initial momentum of the light beam is denoted as M and the nal momentum to each photon is denoted M' in the gure. The momentum change is represented by the red arrow. The force thus induced on the sphere points in the opposite direction, as represented by F. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 xxiv 5.2 Schematic showing a damped driven harmonic oscillator, which illus- trates the motion and the forces on a particle using an applied oscilla- tory laser trap. The solid line denotes the origin of the coordinate sys- tem, the dashed line indicating the location of the bead, and the dotted line the center of the oscillating trap. The equation of motion for the particle is: x(w, t) = −6πη(w)ax(w, m¨ ˙ t)−κ(w)x+kot Aeiwt −kot x(w, t), where kot is the trap stiness of the oscillating beam, A is the amplitude of the oscillatory beam with frequency w, η(w) and κ(w) represent the viscosity and the elasticity of the medium, respectively, and x(w, t) is the bead displacement with respect to the center of the trap (solid line). 61 6.1 Laser tweezers setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.2 Scattering force, gradient force and total trap force on a one micron spherical particle as a function of the displacement of the particle to the trap center. This result is derived by A. Ashkin [9]. . . . . . . . 67 6.3 Measurement of trap force in the linear region of the trap. (a) Power kB T spectral density of a typical bead and a Lorentzian t Sxx (f ) = π 2 β(f 2 +fc2 ) . (b) Trap potential as a function of displacement and a parabolic t. The potential is calculated from the distribution of bead displacement at the same laser intensity as that used in (a). (c) Comparison of trap force obtained from (a), denoted by the solid line, and (b), denoted by the gray dots and a linear t to them shown as the dashed line. . . . 70 xxv 6.4 A picture of the ow cell. The cell is about 15 cm long and 5 cm wide. At the end of the cell are inlet and outlet tubings. The two outlets in the middle are connected to vacuum. A gasket is attached to the bottom of the cell (green) to ensure good isolation from the outside air and prevent leaking of solution. . . . . . . . . . . . . . . . . . . . . . 71 xxvi 6.5 (a) The velocity of a typical bead initially trapped but then dislodged upon application of ow in the ow chamber. (b) The solid circles represent the force on the same bead as in (a) and were t to an expo- nential function to yield y = 1.4 × e−1.5 x (dashed curve). In the linear region, the gray dots (red) within a displacement of about 100 nm are replots from Fig. 6.3c, and the dashed line depicts the expected force prole in this region. 72 xxvii 6.6 An illustration showing the force prole and range of a laser trap. A picture of the trap is shown in red, a probe particle in blue, and the whole force prole in black. The dashed line indicates the position of the beam waist. The dimensions are drawn roughly to scale. The trap force is linear with displacement in the low displacement regime, consistent with experiments. The force drops gradually down to zero beyond the peak when the bead is so far away from the trap center that no laser beams are scattered by the bead. . . . . . . . . . . . . 74 7.1 A schematic diagram of the active microrheology experimental setup [104]. The area enclosed by the dashed lines represents an inverted optical microscope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.2 Moduli for a typical 0.5 mg/ml isotropic F-actin solution (contribution to elasticity due to the trap stiness has been subtracted). Filled symbols represent G' and empty symbols represents G. Black symbols represent data for the direction perpendicular to the sample capillary. Red symbols represent data for the direction parallel to the capillary. The sampling frequency was 20000 Hz and 540000 data points were recorded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 xxviii 7.3 Moduli for a 1 mg/ml actin solution (contribution to elasticity due to the trap stiness has been subtracted). Filled symbols represent G' and empty symbols represents G. Blue symbols represent data for the direction perpendicular to the sample capillary. Green symbols represent data for the direction parallel to the capillary. The sampling frequency was 2000 Hz and 540000 data points were recorded. . . . . 85 7.4 Frequency dependence of moduli for a 4 mg/ml nematic F-actin solu- tion. For the spectra on the left, the bead displacement was measured with the VPT (video particle tracking) method and the mean square displacement was converted to moduli using Generalized Stocks Rela- tion; for the spectrum on the right, the bead positions were tracked by the photodiode. In the overlapped region between the two dash dot lines, the laser tweezers based results are shown in gray symbols. The two methods give consistent results for G0x , G0y , and G00y ; G00x measured by the VPT method, however, is about twice of that measured by laser tweezers based method. Both spectra were measured using 1 µm beads and obtained from averaging over ten beads. The solid straight line shows a scaling law of ω 3/4 . . . . . . . . . . . . . . . . . . . . . . . . 86 xxix 7.5 (A) A representative uorescence image superimposed on a bright eld image, so that both MT bundles (green) and beads (yellow) can be seen. (B) Moduli as a function of frequency for bead 1, showing large dierences between the two directions. . . . . . . . . . . . . . . . . . 90 8.1 (a) The swimming velocity of each of three bacteria as a function of displacement from the center of the trap. (b) The empty symbols represent forces calculated from the velocities in (a), and the dashed curve is a replot from Fig. 6.5b. In the linear region, the gray dots within a displacement of about 100 nm are replots from Fig. 6.3c, and the dashed line depicts the expected force prole in this region. . . . 96 8.2 A schematic drawing which illustrates why the cell body aligns in the direction of the optical axis. Each end of the cell body experiences a trap force against the direction of the laser intensity gradient, shown as F1 and F2 in the gure. The net eect of these two forces is a torque which tends to rotate the cell body back to the optical axis direction. Therefore, the equilibrium position for the cell body is the optical axis direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 xxx 8.3 Selected frames from a movie displaying the escape process of a trapped swarmer cell. The cross indicates the position of the trap center. The white curve on the last image shows the trajectory of the cell. The dark dots indicate the center positions of the cell body over the eight frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.4 (a) Typical voltage signal as a function of time. The voltage is pro- portional to the displacement, projected in each of two orthogonal directions, of the cell body from the trap center. The dark thin curve shows that the total laser intensity was decreased stepwise. The other two curves represent the voltage traces in the two orthogonal directions acquired by the photodiode detector. (b) A magnied view of the volt- age traces right before and after escape of the cell from the trap. (c) Power spectral density of the two voltage traces in (a), showing a wide peak corresponding to the rotation frequencies of the trapped cell. . 100 8.5 Schematic drawing of a cell tilting away from the optical axis. The straight arrow indicates the force generated by the agella motor. The curved arrow indicates the resulting torque which tends to steer the cell towards the horizontal plane. . . . . . . . . . . . . . . . . . . . . 102 xxxi 8.6 Facilitation of Caulobacter adhesion via holdfast by the laser trap. The cross indicates the center of the trap. The laser beam was blocked at later times, thus no cross is shown on the time frame of either 34s or 46s. The inset at the lower right corner of each picture depicts the facilitated adhesion process, in which the box represents the sample chamber, the dark tadpole-like drawing illustrates the cell and the horizontal line represents the position of the focal plane. . . . . . . . . . . . . . . . 103 8.7 An example of DLVO energy of a bacterium interacting with a at glass surface as a function of the distance from the cell to the surface. 106 xxxii Part I Investigation of Mechanism and Characteristics of Microtubule Striated Birefringence Patterns 1 Chapter 1 Introduction to Microtubule (MT) and MT Pattern Formation 1.1 What is microtubule A microtubule (MT) is composed of 13 staggered protolament, built up by longitu- dinally associated α, β -tubulin heterodimers (see Fig. 1.1). The strict alternation of α- and β- subunits results in a longitudinal polarity [55]. Protolaments form sheets at the ends of growing MTs, but curl and peel apart from each other at the ends of shortening MTs, eventually detaching as highly curved oligomers [26]. Microtubules constitute one of the most important components of the eukaryotic cytoskeleton. They participate in many fundamental cellular functions including the maintenance of shape, motility, and signal transmission, and play a determining role 2 3 Figure 1.1: Structure of a microtubule (Adapted from http://www.med.unibs.it). Figure 1.2: Schematic drawing of a mitotic spindle (adapted from www.sparknotes.com). Microtubules aligns and segregates chromosomes during eukaryotic cell division. 4 Figure 1.3: GTP-Cap Model for Microtubule Dynamic Instability. The polymeriza- tion phase is thought to be stabilized by a thin cap of tubulin dimers at the mi- crotubule plus-end with a GTP molecule associated with each beta-tubulin subunit. Stochastic loss of the GTP cap, due to hydrolysis or subunit loss, results in a transi- tion to the depolymerizing phase known as catastrophe. Back transition (rescue) is also observed. Taken from Inoue and Salmon (1995) Mole. Biol. Cell 6:1619-1640. in the organizational changes that occur during the early stages of embryogenesis [30, 21]. Microtubules are a signicant component of brain neuron cells and they make up the mitotic spindles (see Fig. 1.2) that separate the chromosomes during cell division [5]. At the basis of the active role of MTs lies dynamic instability: under the inuence of GTP hydrolysis, a single MT alternates repeatedly between states of prolonged growth (polymerization) and shrinkage, both in a cell and with puried tubulin out- side a cell (see Fig. 1.3). MTs are polymerized from tubulin dimers at physiological temperatures with proper buer conditions including GTP. β tubulin hydrolyzes guanosine triphosphate (GTP) during polymerization [105]. α tubulin also binds GTP, but this GTP is 5 bound in a non-exchangeable manner (at the N-site) and is not hydrolyzed during polymerization [85]. MT polymerization dynamics is inuenced by the addition of taxol. Taxol binds specically to beta-tubulin in the microtubule [47] and causes microtubule bundle formation within the cell. It was found that 1.0 µM taxol induces a lateral aggregation of microtubules which, by 12 to 24 h of treatment, results in the appearance of dense bundles of tightly-packed microtubules [44]. 1.2 Motivation and importance of studies on MT as- sembly and pattern formation Cytoskeletal proteins can form dynamic spatial structures by themselves even in the absence of cellular organizing centers or molecular motors. Solutions of MTs and GTP can generate various dissipative structures including asters, vortices and polygonal networks [66]. The microtubule system could serve as a simple model for studying pattern formation by biomolecules in vitro [59]. The results of these studies may have important relevance and implications for the cellular functions of MTs based on their self-assembly and in connection with their mechanical properties. Of particular relevance here are striped birefringence (light-polarizing) patterns that spontaneously form in MT solutions without motors. Hitt et al. attributed these patterns to the formation of liquid crystalline domains [46]. Tabony, on the 6 other hand, proposed a reactiondiusion based mechanism [88], suggesting that re- actiondiusion could account qualitatively for the concentration variations [89] and the sensitivity of the patterns to gravity and magnetic elds [74]. Similar striated patterns were found in vivo. In particular, an array of parallel wavy microtubules form between the cortical and inner cytoplasm of the vegetal hemisphere in amphib- ian eggs, likely serving as tracks of the microtubule associated motors for cytoplasmic transport [35]. We present results regarding the role of gravity, the structural basis and underlying mechanism for the birefringence stripes and the origin of the spatial concentration oscillations in the striped patterns through a series of microscopic investigations, including phase-contrast, uorescence, and quantitative polarized light microscopy. Based on our observations of MT alignment, bundling, buckling, and the fact that the spatial concentration variation is solely an eect of periodic alignment of bundled MTs, we infer that the pattern formation in our system is mechanical in origin. Ac- cordingly, we present a mechanical buckling model of a microtubule bundle within an elastic network formed by other similarly aligned and buckling bundles and unaligned MTs to explain the shape, wavelength, critical buckling force and other characteris- tics of the patterns such as retardance evolution, bundle growth rate and packing geometry. Moreover, we studied the eect of polymer molecules such as Polyethylene Glycol (PEG, MW=35 kDa) on microtubule bundling and pattern formation. The 7 result regarding the modication of the pattern by PEG yields a better understand- ing of the mechanism of formation and characteristics of MT bundles in the striped patterns. Chapter 2 Experimental Methods 2.1 Experimental chamber Intracellular organization has to do with the conned geometry of cells. To simulate the closed environment of cells, we use rectangular cuvettes which have dimensions of 40 ×10 ×1 mm made of quartz (International Crystals, Oklahoma City, OK) or 50× 8 ×0.4 mm made of glass (Vitrocom, New Bedford, MA). 2.2 Purication of tubulin and polymerization of MTs Tubulin was puried from fresh bovine brains by two cycles of polymerization and depolymerization, followed by cation exchange chromatography on a phosphocellulose column following standard protocols [99, 103]. Tubulin-containing fractions were pooled, concentrated by additional cycle of assembly and disassembly, homogenized 8 9 in PM buer (0.1 M PIPES, 2 mM MgCl2, 1 mM EGTA, pH 6.9) with 0.5 mM GTP, disbursed into aliquots, frozen in liquid nitrogen, and stored at  80 °C. Immediately before use, tubulin solutions were thawed and then centrifuged at 1,800 g for 5 min at 4°C to remove small amounts of aggregates. For most of our samples, the GTP concentration was 2 mM unless otherwise specied. Samples were degassed (after the addition of GTP) in cuvettes at 4°C to prevent air bubble for- mation that would otherwise occur during the increase in temperature to 37°C. The cuvettes were incubated on ice for 10 min before inducing polymerization. 2.3 Kinetics measurement Microtubule assembly was initiated by adding GTP into a tubulin solution (4 °C) and lling the mixture into a rectangular cuvette, and then inserted the cuvette into holder in a circulating water-warmed Shimadzu spectrophotometer. The air temper- ature within the Shimadzu chamber was about 34  35 °C. Kinetics measurements were performed immediately after by recording optical density (i.e., absorbance) at 350 nm every two seconds for 1500 seconds (25 mins). What is measured is dimin- ished transmittance of light due to scattering o objects of macromolecular particles. As tubulin polymerizes into microtubules, incident light scatters more the more mi- crotubules (approximately 0.5 µm or longer) are polymerized. Assuming that Beer's law holds, I −log( ) = kc I0 10 , where I0 is the incident light intensity and I is the outgoing light intensity. k is extinction constant and c is the concentration of MTs formed. 2.4 Aligning MTs with static magnetic elds The application of static magnetic elds has been a non-intrusive mechanical method to align biological samples [16, 107, 94, 29]. In particular, the alignment of MTs in a magnetic eld of a few Tesla has been shown by Glade et al. [39]. Theoretical estimation shows that the minimum eld strength needed to align a 5 µm MT parallel to the magnetic eld direction is about 7.6 Tesla [16]. For MTs longer than 5 µm, the required eld strength is even smaller. The magnetic eld in our experiments was produced by a super-conducting magnet system (American Magnetics, Oak Ridge, TN) with a room temperature bore. The diameter of the bore is 11 mm. Pre-cooled tubulin solutions in glass cuvettes were placed in a 9 Tesla vertical magnetic eld oriented parallel to the long axis of the cuvettes. The temperature of the bore was ◦ pre-equilibrated to 37 C by circulating warm air through it from above. The sample ◦ ◦ temperature, measured with a thermometer, rose from 0 C to 37 C in about 100 ◦ s. Observations and measurements were made at either room temperature or 30 C, when specied, after the sample was removed from the magnet. 11 2.4.1 Magnetic torque on a single MT The magnetic energy of a single MT oriented at an angle θ with the direction of a magnetic eld B is: L∆χ 2 ∆E = − B cos2 θ (2.1) 2µ0 Here L is the length of the MT, ∆χ is the diamagnetic anisotropy of MT. Therefore the torque induced by the magnetic eld is: ∂∆E τB = ∂θ L∆χ 2 = B sin 2θ (2.2) 2µ0 2.4.2 Time needed to align a MT in a magnetic eld The equation governing the rotation of a MT in a magetic eld is: dθ β = −τB , dt 2π ηL3 Here β= 3 log(L/d) is the rotational drag coecient and d is the diameter of a MT. Therefore, 2π ηL3 dθ L∆χ 2 = − B sin 2θ 3 log(L/d) dt 2µ0 The above equation yields: 12 µ0 β t = (ln tan θ − ln tan θ0 ) L∆χB 2 µ0 2π ηL2 = (ln tan θ − ln tan θ0 ) ∆χB 2 3 ln(L/D) ∝ L2 , where θ0 and θ are the initial and nal angle between the MT and the magnetic eld direction. Numerically, the time needed to rotate a 10 µm long MT from θ0 = 45◦ to θ = 10◦ in a 9 Tesla magnetic eld is about 47 s. 2.5 Fluorescence and phase-contrast microscopy Microscopic structures of self-organized MT pattern inside cuvettes were examined with uorescence and phase-contrast microscopy at room temperature on a Nikon ECLIPSE E800 microscope. Images of 12-bit depth were collected by a 1,392×1,040- pixel Coolsnap digital camera (Roper Scientic, Trenton, NJ) driven by META- MORPH imaging software (Universal Imaging, Downingtown, PA). For uorescence imaging, MTs were labeled to a 3.5% stoichiometry in the ratio of total number of Oregon green-conjugated taxol (Molecular Probes) molecules to tubulin dimers. Blue light (wavelength 480 nm) was used for excitation, and green light emitted by taxol-Oregon green (peak wavelength 532 nm) was measured. 13 2.6 Birefringence measurements The birefringence of objects is a sensitive indicator of structural order at dimensions ner than the wavelength of light. Birefringence can vary in time and space, revealing the subtle changes in ordered arrangements. One easy and quick method to examine sample birefringence is to place the cuvette between two sheets of Polaroid (Edmund ◦ Optics, Blackwood, NJ) oriented with their axes of transmission at 90 (extinction 99.98%) and illuminated by a light box (Hall Productions, Grover Beach, CA). 8-bit depth images were recorded by a CCD camera (Sony, XCD-SX900, NY) driven by Fire-I software at a resolution of 1280 × 1024. More quantitative birefringence measurements were performed using a Nikon ECLIPSE E800 microscope equipped with a PolScope package (CRI, Cambridge, MA) and a standard CCD camera (MTI 300RC, Michigan, IN) with 640 × 480 resolution. Pol- scope technique is a sensitive and reliable tool for quantitative measurements of mi- croscopic distributions of birefringence of biological samples [84]. The technique uses elliptically polarized illumination beam which is produced by a combination of a linear polarizer and two liquid crystal retarders. The polarization property of the illumina- tion beam changes after going through the sample. The nal polarization was probed by a circular analyzer. The intensity distribution of light on the image plane depends on the retardation of the liquid crystal retarders and birefringence of the sample. The functional dependence can be calculated by multiplying the Jones matrix of the illumination beam and the matrixes of the optical components including the sample 14 that the beam goes through. Four settings of the liquid crystal retarders were used to calculate the corresponding intensity distribution. Solving the four functions re- lating the retarder settings and the intensity distribution yields the retardance and slow axis distribution of the sample. Background retardance caused by other optical components is subtracted [71]. 2.7 MT bundle size measurement using PolScope MT bundle size was calculated from the birefringence of the bundle as measured using PolScope. Retardance images were analyzed using Matlab. Specically, a line scan was taken on the cross section of a MT bundle and the intensity along the cross section was integrated to yield the retardance area. Given that the integrated retardance area of a single MT is A = 7.5 nm2 [72, 51], the number of MTs along the cross section of the bundle can be determined. 2.8 Sedimentation and quantitative uorescence as- say This assay was performed to test the contribution of free uorescent taxol and free tubulin associated with taxol to the total uorescence signal. First, the uorescence of a solution of uorescent taxol labeled (with a stoichiometry of 3.5% in the ratio of the total number of taxol molecules to that of tubulin dimers) MTs was measured 15 using a uorescence spectrometer (PerkinElmer, Wellesley, MA). Then the solution was centrifuged at 280,000g for 30 min to sediment MTs. The uorescence of the supernatant was subsequently measured. Chapter 3 Properties of Microtubule Striped Birefringence Patterns 3.1 Macroscopic appearance of birefringence patterns When a sample of microtubules is inserted between two polarizers that have their axes of polarization at 90° to each other, the fraction of light which can pass through depends on the MT polarization. Thus, the axis of alignment of MTs can be mapped out based on the distribution of light passing through. Fig. 3.1 shows a birefringence image taken through crossed polarizers. The dimen- sions of the cuvette are 40 x 10 x 1 mm. Fig. 3.2 shows a similar pattern formed by a MT solution which was exposed to a 9T magnetic eld for 5 mim after initiation of 16 17 45 min after initialization of polymerization Figure 3.1: MT striped birefringence pattern observed between crossed polarizers. o o The axis of the cuvette was at 0 (left) and 45 (right) relative to that of the polar- ization directions of the polarizers, respectively. The dimensions of the cuvette are 40 x 10 x 1 mm. crossed polarizers 5 m g/m l T ubu lin solution 5 m in in 9 T esla m agnetic field cuvette size: 0.4 m m thick, 8 m m w id e Figure 3.2: Pattern development of a polymerizing 5 mg/ml MT solution following initial alignment by a 9 Tesla vertical static magnetic eld for 5 min. The images were taken between crossed polarizers with the long axis of the cuvette (8 mm wide, 0.4 ◦ mm thick) at 45 with respect to the polarizing directions of the polarizers. (A) Image after 5 min of polymerization in the magnetic eld. Notice that the solution appears brighter than the background region outside the meniscus, which has no birefringence. ◦ Rotating the cuvette by 45 changes the image to dark (data not shown), indicating that MTs are aligned in the magnetic eld direction. (B) The same sample imaged 30 min after initiation of polymerization, showing formation of stripes which are aligned predominantly in the direction perpendicular to the direction of initial bulk alignment. 18 1h 2h 3 .5 h 5 .5 h g Figure 3.3: Time evolution of a MT pattern. The images were taken between crossed ◦ polarizers with the long axis of the cuvette (8 mm wide, 0.4 mm thick) at 45 with respect to the polarizing directions of the polarizers. The arrow indicates the direction of gravity. 19 Figure 3.4: Time series of a MT pattern formation during the initial stage. Dye particles were added to the tubulin solution before polymerization. polymerization. Alternating bright and dark regions were observed, indicating undu- lating microtubule orientations. Dynamic in nature, the overall shape of the pattern does not change much over the course of 6 hours (see Fig. 3.3), it only becomes more sharp and detailed. 3.1.1 Role of convective ow and gravity in pattern formation Existence of convective ow was investigated by adding Phenol Red dye in the bottom of a cuvette before tubulin solution was added. Upon non-uniform increase of solution temperature, convection was observable by tracing the red dye particles. Fig. 3.4 displays a series of images from a movie where the motion of dye particles following the ow was apparent. A further evidence of ow induced alignment and subsequent pattern formation is that similar wave-like patterns (see Fig. 3.5) form in narrow 20 10X fluorescence image, 6hours after shear flow Figure 3.5: Wave-like patterns formed in capillaries where shear ow was purposely induced. The capillary is 500 µm wide. capillaries in which shear ow was purposely induced through injection of solution. Stripes were less apparent when initiating MT polymerization in a upright cuvette which was warmed up uniformly, for example, when immersing the cuvette in a 37°C water bath. On the other hand, strips did form when warming a at cuvette nonuni- formly (data not shown). These control experiments demonstrate that gravity does not directly play a role in the formation of striped patterns, but convective ows do. No strips formed in experiments performed in space [90] because no convection was available to align MTs. 3.2 MT bundle formation The sequence of phase contrast microscopy images shown in Fig. 3.6 reveals the mor- phology of the MTs inside striped patterns. At 10 min, striations with a gradual 21 Figure 3.6: Successive frames of a typical phase-contrast movie showing buckling of MT bundles. The contrast of the images here was enhanced for better visualization. 8 mg/ml tubulin solution in a 40 × 10 × 1 mm cuvette was prepared as described in Materials and Methods. The sample was subjected to convective ow (induced by asymmetrical thermal contacts between the two edges of the cuvette with a water bath-warmed aluminum holder) for the rst 9 min and then the cuvette was laid at ◦ on the microscope stage at 30 C. undulation are apparent. This undulation becomes stronger with time until its am- plitude is comparable to its spatial period. We identify the striations as bundled MTs since single MTs are unresolvable with this technique. Bundles of MTs are clearly recognizable in uorescence and PolScope images of samples taken out from cuvettes which contain striped patterns, as shown in Fig. 3.8. MT clouds instead of bundles were observed in samples in which the stripes did not form. This result suggests that MT bundling is necessary for the formation of the striped patterns. MT bundles observed at higher magnication using uorescence microscopy are 22 A 80 μm B 30 μm Figure 3.7: MT bundles detected by both birefringence and uorescence measure- ments. MTs polymerized from a 5 mg/ml tubulin solution in glass cuvettes were taken out and then applied on a coverslip and covered with a glass slide. (A) Results of the PolScope measurement. Brightness indicates the magnitude of retardance and pins represent the slow axis directions which are consistent with the orientations of MT bundles. (B) Fluorescence image of the rectangular region as indicated in A. 23 20 µm Figure 3.8: MT bundles observed using high magnication uorescence microscopy. MTs polymerized from a 5 mg/ml tubulin solution in glass cuvettes were taken out and then applied on a coverslip and covered with a glass slide. Figure 3.9: A typical confocal image of MT bundles (green). MTs were polymerized in glass capillaries and labeled with taxol conjugated Oregon Green. 24 shown in Fig. 3.7. A typical confocal image of MT bundles is shown in Fig. 3.9, in which MT bundles are green. Although these bundles appear as loosely packed aggregates of parallel microtubules (because bundles were disturbed during pipeting and subsequent pressing on the cover slide), it is highly likely that microtubule bundles inside striped patterns are much more tightly packed, as demonstrated later in the discussion of the mechanical buckling model. It has been reported that bundling transitions of lamentous biopolymers such as DNA, F-actin and MTs can be induced by either electrostatic or steric interac- tion [91]. The electrostatic interaction stems from the polyelectrolyte nature of these biopolymers [91], which gives rise to nonspecic binding by ligands carrying oppo- site charges including counterions. At suciently high concentration of counterions of appropriate valency, typically divalent or higher, a net attractive interaction can occur, leading to the subsequent lateral aggregation. The steric interaction refers to a distinct physical mechanism known also as depletion attraction to colloidal scien- tists or macromolecular crowding in the biochemistry literature [58]. It has been theoretically predicted and experimentally shown that rigid laments coexisting with a crowded solution of inert polymers or globular proteins may spontaneously coalesce into bundles [58, 50]. Both interactions discussed above are likely relevant to the bundle formation of MTs in this study. MTs are known to be highly negatively charged, and are sur- rounded by counterions, particularly the divalent Mg 2+ ions existing in millimolar 25 concentration. The counterions signicantly screen the negatively charged MTs, re- ducing the mutual repulsion between them. Theoretical work over the past decades has predicted attractive interactions due to correlations of counterions shared by par- allely aligned rods of like charge [73, 83]. Experimentally, MTs have been observed to form bundles in the presence of divalent and multivalent cations [92, 67]. As an additional factor, unpolymerized tubulin dimers, oligomers and even short MTs may serve as "inert" macromolecules, which facilitate the lateral aggregation of parallely aligned MTs via the steric interaction. Indeed, MTs have been shown to form bundles of dierent morphologies in the presence of inert macromolecules [68]. We speculate that MT bundle formation in our system is due to the combined eects of counterion induced attraction and depletion force. The 2 mM divalent counterion Mg 2+ present in the MT solutions may be insucient to induce substantial bundling alone. Combining its eect with the depletion attraction, however, and given the fact that the polymerizing MTs are concurrently being aligned by external forces, bundling becomes the likely outcome. In fact, experimental evidence of spontaneous bundling of MTs has been given by Hitt et al. [46]. In this study, we conrm the previous report and show that bundling of MTs is a key step towards developing the striated patterns. 26 3.3 Small angle x-ray scattering experiments and data analysis Small angle x-ray scattering (SAXS) is a technique where the elastic scattering of x- rays (wavelength 0.1 to 0.2 nm) by a sample is recorded at very low angles (typically 0.1o - 10o ). This angular range contains information about the shape and size of macromolecules, characteristic distances of partially ordered materials, pore sizes, and other data. SAXS is capable of delivering structural information of macromolecules between 5 and 25 nm, of repeat distances in partially ordered systems of up to 150 nm [40]. In the case of biological macromolecules such as proteins, the advantage of SAXS is that a crystalline sample is not needed. However, due to the random orientation of dissolved or partially ordered molecules, the spatial averaging leads to a loss of information in SAXS. In this study, we used SAXS to study the scattering of mi- crotubule bundles. The x-ray source was synchrotron beam at Brookhaven National Laboratory. To prepare sample for measurements, 5 mg/ml tubulin was polymerized in eppendorf tubes and injected into a 1.5 mm diameter cylindrical x-ray tube. The wavelength of x-ray is 0.776 angstrom, and the data acquisition time is 10 minutes. In a typical SAXS experiment, a monochromatic beam of x-rays is brought to a MT sample from which some of the x-rays scatter, while most simply go through the sample without interacting with it. The scattered x-rays form a scattering pattern 27 which is then detected by a detector positioned behind the sample perpendicular to the direction of the primary beam that initially hit the sample. The scattered intensity I(q) is recorded as a function of momentum transfer q (q=4π sinθ/λ, where 2θ is the angle between the incident and scattered beam. Fig. 3.10 shows a typical scattering pattern, the brightness indicates the scatter- ing intensity. The random positions and orientations of MTs result in an isotropic intensity distribution. Fig. 3.11 plots the scattering vector q versus the logarithm of the scattering intensity (azimuthally averaged). The scattering pattern is close to that of a dilute solution of MTs. The peaks correspond to the Fourier transform of the MT cylinder if we compare Fig. 3.11 with Fig. 3.12. We do not see peaks corresponding to packing of MTs in bundles if we compare Fig. 3.11 with Fig. 3.13 and other scattering plots of MT bundles in the literature [67, 68, 69]. Therefore, we speculate that most MTs are dispersed in the samples tested and the scattering from small fraction of bundles was smeared by the scattering from single MTs. The low extent of bundle formation here as compared with that in striated patterns was probably due to lack of alignment factors such as convection and magnetic elds during polymerization. 28 Figure 3.10: Scattering pattern from a SAXS scan. The scattering pattern contains the information on the structure of the sample. The solvent scattering has been subtracted. Dierent rings around the axis of the initial incident beam represent dierent scattering angles θ. The scattering intensity I(q) at each scattering angle was averaged over the region between the two red lines. The brightest region in the center is where the strong main beam (containing unscattered beam) hit. 8.5 8.0 7.5 log (I) (a.u.) 7.0 6.5 6.0 5.5 5.0 4.5 0.0 0.05 0.1 0.15 0.2 0.25 q (inverse angstrom) Figure 3.11: Plot of logarithm of scattering intensity log(I) versus scattering vector q. The peak positions are at 0.0046, 0.028, 0.057, 0.084 inverse angstrom, respectively. 29 Figure 3.12: Scattering from oriented solution of hydrated MTs [16]. The value of q for the rst peak is at 0.027 inverse angstrom, the second peak is at 0.055 inverse angstrom, the third peak is at 0.084 inverse angstrom. All these three peaks correspond to Fourier transform of the MT cylinder. Figure 3.13: Theoretical interparticle interference function [62]. d and f corresponds to arrays of nite size with perfect hexagonal symmetry corresponding to regular bundles of 19 microtubules; e and g corresponds to nite paracrystals in two dimensions corresponding to bundles of microtubules (about 20 MTs per bundle) with deviations from hexagonal arrangement. 30 3.4 Birefringence measurement, phase contrast and uorescence imaging of striated patterns We use birefringence measurement, phase contrast and uorescence imaging to study the microscopic characteristics of striated patterns. Fig. 3.14A, Fig. 3.15A, Fig. 3.15B shows a typical PolScope, phase contrast and uorescence image of the striated pattern respectively. Both the solution retardance magnitude and the slow axis orientation of MTs vary in space (see Fig. 3.14A). The retardance values and the MT slow axis orientations along the white line in Fig. 3.14A are plotted in Fig. 3.14B and Fig. 3.14C, respec- tively. Note that the retardance is higher in the sloped regions and lower in the peak and valley regions of the waves. Similar to the retardance, the uorescence inten- sity undulates in space as shown in Fig. 3.15B. The wave-like striations in the phase contrast image are due to MT bundles (single MTs are not resolvable using phase con- trast microscopy) undulating in a wave-like manner, which give rise to macroscopic birefringence stripes between crossed polarizers. 3.4.1 Quantitative analysis of periodic variations in birefrin- gence and uorescene intensity Concurrent undulations in the retardance and uorescence intensity of labeled MTs are apparent as mentioned in the previous paragraph. These undulations have half 31 A y 300 µm x B 20 Retardance (nm) 16 12 8 0 500 x (µm) 1000 1500 C 90 45 ϕ 0 -45 -90 0 500 1000 1500 x (µm) D E 20 Retardance (nm) ϕ S 15 D ϕ 10 5 0 0 1 2 3 1 + tan 2 ϕ Figure 3.14: Quantitative analysis of the striped patterns based on birefringence measurements [57]. (A) PolScope measurement of a typical sample self-organized in 3.3 mg/ml tubulin solution (taxol-Oregon Green added for uorescence imaging performed on the same sample). The solution in a 0.4 mm thick cuvette was initially exposed to a 9 Tesla magnetic eld for 30 min. The measurement was done after the pattern was stable. The brightness indicates the magnitude of retardance and the pins show slow axis orientations. (B) Retardance values along the white line in p A (blue curve) and t (red curve) using ∆(x) = ∆0 1 + tan2 ϕ(x). The x axis starts from the left end of the white line in A. The slow axis orientations ϕ are measured values as shown in C. The only tting parameter is ∆0 . (C) Slow axis orientations along the same line. (D) Schematic showing that the packing density is p proportional to 1 + tan2 ϕ for nested MT bundles (each curve represents p a bundle). (E) Retardance along the white line in A and t proportionally to 1 + tan2 ϕ, where ϕ is the slow axis direction at each corresponding position. This gure is taken from [57]. 32 A y 200 μm x B 200 μm C Fluorescence Intensity 600 550 500 450 400 0 250 x (µm) 500 750 D 600 Fluorescence Intensity 400 200 0 0.0 0.5 1.0 1.5 2.0 1 + tan 2 ϕ Figure 3.15: Quantitative analysis of the striped patterns based on uorescence mea- surement [57]. (A) Phase contrast image (contrast enhanced) of the same region as in Fig. 3.14 but with a smaller eld of view. Blue points are manually picked up along a MT bundle. The red curve is a t to the shape of the bundle using the sum of three sinusoidal functions. (B) Fluorescence image of the same region, the red curve is the same as in A. (C) The blue curve is a plot of the uorescence intensity along the red curve in B as a function of the horizontal axis, x. The red curve is a t using p I(x) = Ib + I0 1 + tan2 ϕ(x), witch yields Ib = 299 and I0 = 143. Here tanϕ(x) corresponds to the calculated slope of the contour ofpthe bundle based on the t to its shape. (D) Linear t of uorescence intensity to 1 + tan2 ϕ using data from C. The non-zero intercept of the line with the ordinate implies a background uorescence caused by dispersed MTs. This gure is taken from [57]. 33 the spatial period of the wavy bundles and indicate how the MT density varies. The nesting of buckled bundles accounts for the observed retardance and uo- rescence intensity variations. As shown schematically in Fig. 3.14D, bundles of MTs buckle into a wave-like shape and nest with their neighbors. We assume that the transverse bundle displacement, S, is independent of the horizontal position, x. The local packing density of bundles (number of bundles per µm2 ), P , however, depends on x through ϕ(x) as: 1 1 1p P ∝ = = 1 + tan2 ϕ(x) (3.1) D S cos ϕ S where ϕ is the local angle of the MT bundles measured relative to the original align- ment direction. As the retardance magnitude is proportional to the packing density p of aligned laments [43, 81], it is expected to follow ∆(x) = ∆0 1 + tan2 ϕ(x). To p test this expectation, the measured ∆ in Fig. 3.14B is plotted against 1 + tan2 ϕ in Fig. 3.14E. These data are consistent with the expected proportional relationship and a t yields ∆0 =7.44 nm. The nesting of buckled bundles also accounts for the coexisting uorescence inten- sity variations. The uorescence intensity contributed by MTs aligned in the bundles p is also expected to follow I∝ 1 + tan2 ϕ. Unlike the retardance, however, there is an additional background contribution, Ib , which is due to dispersed MTs. These MTs are most likely randomly oriented as they do not contribute to birefringence. To test these expectations, we use a typical phase contrast image (Fig. 3.15A) to follow the contour of a bundle (i.e., to determine ϕ(x) along the bundle). The blue points were 34 picked along a discernible bundle and their positions were t to a simple Fourier ex- pansion with three sinusoidal terms. The measured uorescence intensities along the red tted curve in Fig. 3.15B are plotted as the blue curve in Fig. 3.15C versus the hor- p izontal position, x. The red curve in Fig. 3.15C is a t to I(x) = Ib +I0 1 + tan2 ϕ(x) which yields Ib = 299 and I0 = 143 (see also Fig. 3.15D), where I(x) is the total u- orescence intensity and tan ϕ(x) is the calculated corresponding slope of the bundle based on the t to its shape. 3.5 Decomposition of the pattern structure into MT bundles and dispersed MTs in the network It is known that taxol binds tightly and specically to MTs (see, for example, [47]). Thus the measured uorescence signal mainly comes from MTs incorporated into bundles and in the surrounding network. To validate this assertion, we centrifuged a solution of uorescent taxol labeled MTs and measured the uorescence signal of the supernatant. The signal after sedimenting MTs was less than 3% of the original solution (data not shown), conrming that the contribution to the uorescence signal from free uorescent taxol and any uorescent taxol bound to either tubulin dimers or short fragments of MTs that do not sediment is negligible. Using the background and the average total uorescence intensity, Ib and I¯, re- spectively, we calculated the fraction, η , of MTs incorporated into the nested bundles 35 after the pattern became stable. For the sample presented in Fig. 3.15, I¯ = 485 (see Fig. 3.15C), I¯ − Ib 485 − 299 η= ¯ = = 38% (3.2) I 485 where I¯ − Ib is the contribution from MTs incorporated into the bundles. These calculations demonstrate that only a fraction of MTs forms the undulating bundles. The other fraction of MTs exist in a non-birefringent, isotropic network based on the fact that there is no background retardance (see Fig. 3.14E). 3.6 Estimation of MT bundle size and bending en- ergy based on retardance measurement We estimate the average number of MTs in the cross section of a bundle from the retardance image shown in Fig. 3.8A. The retardance signal of a single MT has a peak of 0.07 nm and a spread of 0.17 µm [72, 51]. The average retardance in Fig. 3.8A is about 2 nm. So, the number of MTs in Fig. 3.8B which has a width of about 150 µm can be obtained as follows: 2nm × 150µm NM T = = 25, 210 0.07nm × 0.17µm 36 The total number of bundles in Fig. B is estimated to be about 100, so each bundle contains NM T 25, 210 n2 = = = 252 M T s/bundle 100 100 The bundle radius can be calculated as follows: πR2 = 250 π(12nm)2 =⇒ R = 16 × 12nm =⇒ R = 192nm Assuming the shape of the bundle is y = aλ sin 2πx λ (a = 1 2 ), the bending energy of all the MTs in a 5 mg/ml solution in a cuvette with sizes 8mm × 0.4mm × 40mm is: 37 L EM T s = × EM T,per wavelength Lper wavelength k λ 2 Z L = × ρ ds Lper wavelength 2 0 k λ y 002 p Z L = Rλp × 1 + y 02 dx 1 + y 02 dx 2 0 (1 + y 02 )3 0 Z 1 L k 8a2 π 4 sin2 2πt = × × {R 1 √ × dt} λ λ 1 + 4a 2 π 2 cos2 2πtdt 0 (1 + 4a2 π 2 cos2 2πt)5/2 0 L π = Constant × 2 × ER4 λ 4 7 1.64 × 10 × 40mm π = 11.1 × 2 × × 1.2 × 109 pa × (12nm)4 (600µm) 4 = 3.95 × 10−10 J = 9.55 × 1010 KB T Here L is the length of the cuvette, 1.64 × 107 is the total number of MTs in a 5 mg/ml solution in the cross section of a cuvette with sizes 8mm × 0.4mm × 40mm (assuming each MT has a length of L and the critical concentration of polymerization is 1 mg/ml). On the other hand the energy released by GPT hydrolysis can be estimated as follows: EGT P = 2mM × 8mm × 0.4mm × 40mm × 6.02 × 1023 × 10KB T = 1.6 × 1018 KB T Therefore, energy provided by GTP hydrolysis is enough for buckling to occur. 38 3.7 Time evolution of retardance, bundle length and buckling wavelength Time lapse phase contrast microscopy reveals that MT bundles elongate uniformly along their contour during buckling, which is consistent with polymerization occurring uniformly along the bundles. Bundle elongation is illustrated in the phase contrast images Fig. 3.16(a) and Fig. 3.16(b), showing a xed region taken 12 and 100 min after polymerization initiation, respectively. The three white curves in each image are computer generated traces of bundle contours that extend between selected ducial marks. The ducial marks are visible as dark spots. To generate the white curves, we presumed that the bundles followed the striations and traced the stripes between the ducial marks. The marks were tracked by MetaMorph (Universal Imaging, West Chester, PA). We determined the local striation orientation at each pixel by calculat- ing a Fast Fourier Transform (FFT) of the area around the pixel, shown, for example, in Fig. 3.16(c). The FFT appeared as an elongated spot oriented perpendicular to the striation direction [Fig. 3.16(d)]. The radially integrated FFT intensity has a peak at a specic azimuthal angle [Fig. 3.16(e)] that is perpendicular to the stria- tion orientation. In this way, the lengths of three segments along a MT bundle were recorded every 30 seconds and plotted in Fig. 3.16(f ), (g) and (h). The normalized lengths of these three segments grew at nearly the same, constant rate, shown in Fig. 3.16(i), implying that the MT bundles elongate uniformly along their contour 39 (a) 1 2 3 46 mins 1 (b) 2 3 100 µm 106 mins (c) (d) FFT Intensity (e) 4.4 4.2 θ 0 θ 180 (f) (g) (h) (i) 90 320 L ( µm) 1 2 230 3 1.1 L/L0 85 1,2,3 300 220 1.05 1 46 106 46 106 46 106 46 106 t (mins) t (mins) t (mins) t (mins) Figure 3.16: Illustration and measurements of the uniform elongation of MT bun- dles [1]. (a),(b) Phase contrast images of a sample region, showing progression of the pattern over 1 h. MT bundles are discerned by the thin striations. Segments 1 through 3 are adjacent pieces of a contour followed by bundles. The segment ends are dened by ducial marks. (c) Magnied view of the region denoted by the white box in (b), showing an encircled ducial mark. (d) Fast Fourier Transform (FFT) of (c). (e) The radially averaged FFT intensity plotted versus the azimuthal angle θ and t using a Gaussian function. The local bundle orientation is orthogonal to the angle at which the Gaussian t peaks. (f )-(h) Length of segments 1 (f ), 2 (g), 3 (h) as a function of time. (i) Lengths of the three segments as a function of time, normalized to their lengths at 46 min. This gure is taken from [42]. 40 10 20 30 40 (nm) (a) 12 mins (b) 100 mins 200 µm 200 µ m (c) 12 mins 200 (d) 1.6 y ( µm) L 1.4 0 L0 1.2 −200 1 0 1000 12 50 100 (e) x ( µm) (f) t (mins) 25 800 600 λ ( µm) ∆ 20 400 (nm) 15 200 0 1 1.2 1.4 1.6 12 50 100 L/ L 0 t (mins) Figure 3.17: Time evolution of a MT pattern obtained by measuring the retar- dance and slow axis of the sample using a PolScope imaging system [2]. (a),(b) Retardance images of a sample region at 12 and 100 min of self-organization, re- spectively. The color bar shows the retardance magnitude scale and the green pins provide the slow axis orientation. The straight white lines represent the slow axis line scan position.(c) Slow axis line scan (black) and the tted slow axis orienta- 2π 2π tion ϕ(x) = arctan{A cos[ (x + x0 )]} (red) at 100 min. (d) The dominant buckling λ λ wavelength λ, obtained from the tted shapes of the bundle at individual time points. (e) The length evolution of the tted bundle contour. L0 = 1544 µm is the initial unbuckled length of the bundle. The segment before the arrow designates the latent period prior to the onset of the buckling. (f ) The magnitude of the retardance aver- aged over the line as shown in (a) versus the normalized length L/L0 . This gure is taken from [42]. 41 instead of growing solely at their ends. It further suggests that the bundles elongate through polymerization of their constituent MTs, which start and end at random places along a bundle. The uniform growth of all MTs within a bundle justies a uniform compressional force throughout the bundle during buckling. Additional quantitative information about the microscopic picture of the buckling is gained through time-lapse birefringence measurements. PolScope images, taken sequentially at a xed sample region [2], yielded the time evolution at each pixel of both the retardance (∆ ≡ birefringence ×h, where h is the sample thickness) and slow axis direction [ϕ(x), orientation of MT bundles]. Two representative PolScope images of a single region taken at dierent stages of self-organization are shown in Fig. 6.2(a) and 6.2(b). The slow axis variation, ϕ(x), along the white lines in Fig. 6.2(a) and 2π 2π 6.2(b) can be t to ϕ(x) = arctan{A cos[ (x + x0 )]}, indicating that the bundle λ λ follows ξ(x) = A sin[ 2π λ (x + x0 )] with a single wavelength λ, buckling amplitude A, and oset x0 [Fig. 6.2(c)]. The resultant wavelength, λ ≈ 600 µm, is plotted in Fig. 6.2(d). The normalized contour length calculated from the ts, L(t)/L0 , grew nearly linearly with time at a normalized rate of ˙ L(t)/L0 ≈ 1% per min [Fig. 6.2(e)]. Simultaneously, the retardance magnitude averaged over the white line in Fig. 6.2(a) increased roughly in proportion to L(t)/L0 [Fig. 6.2(f )]. Based on the nesting model we proposed earlier and assuming that neighboring MT bundles do not coalesce, the average retardance goes as ∆(t) ∼ δ × n(t)L(t)/L0 [57, 72], where n(t) is the number of MTs in the cross section of a bundle and δ is the retardance of a single MT. 42 Therefore, the linear relation between ∆(t) and L(t)/L0 implies that n(t) remains constant throughout buckling. Thus, the elongation of MT bundles occurs through the polymerization of MTs within the bundles and does not involve the incorporation of new MTs to existing bundles. Chapter 4 Mechanism of Striped Pattern Formation and a Mechanical Buckling Model 4.1 Mechanism of striped birefringence pattern for- mation We propose a two-stage mechanism for the stripe formation: polymerizing MTs align uniformly along the direction of external magnetic elds or convective ow and form bundles in the early stage; the bundles grow uniformly due to the elongation of con- stituent MTs and eventually buckle and nest into a wave-like shape in the later stage. The two-stage mechanism for the stripe formation not only involves the chemical and 43 44 biological reaction of MT polymerization and bundle formation, but also incorporates a mechanical component of buckling of growing MT bundles. This mechanism might account for some large scale phenomena in biological morphogenesis and provide a basis for further experimental and theoretical work. 4.2 Estimation of buckling wavelength based on min- imization of energy We estimate the characteristic wavelength of buckling by minimizing the sum of bending energy of the bundles, the elastic energy of the MT network surrounding the bundles and the compressional energy of the bundles. We assume that buckling arises once the compressional strain of a growing MT bundle exceeds a critical value and the characteristic wavelength is selected at the onset of the buckling instability with the critical strain. We consider MT bundles as homogeneous elastic rods, the buckling of which costs bending energy. The MTs incorporated into bundles are not the only population of MTs in the system (see Eq. 3.2), but there exists another population of MTs which form an elastic MT network. Thus, buckling of the bundles involves an elastic deformation of the surrounding MT network and induces an elastic energy to the network. We assume that the elastic restoring force applied by the network on the bundle per unit bundle length, F, is proportional to the displacement of the bundle in the transverse direction, ξ(x), as: F (x) = α ξ(x), where α is the elastic constant 45 of the network per unit length along the bundle. The actual mode of buckling is selected by minimizing the sum of the bundle bending and compressional energy and the network elastic energy. The bending energy scales approximately as: Z λ κ d Ebending = ρ2 ds 2 0 λ κ 8π 4 δ 2 d ≈ 2 λ3 λ δ2d = ER4 π 5 λ4 where κ is the bending stiness of the bundle, E is the Young's modulus of the bundle, d is the end to end distance of the bundle, R is the radius of the bundle and δ is the amplitude of buckling, which is much smaller than λ at the onset of buckling. The elastic energy of the network scales as: 1 δ2 α Eelastic = α d = δ 2 d 2 2 4 i.e., Eelastic ∝ α δ 2 . The bundle compressional energy is calculated as follws: 1 1 EA 2 2 1 Ecompression = kbundle 2 L2 =  L = EA2 L 2 2 L 2 where kbundle is the spring constant of the bundle,  is the strain, L is the bundle length and A is the cross sectional area of the bundle. The constraint of a xed contour length of the bundle dictates that 46 Figure 4.1: Schematic drawing to illustrate the elastic buckling model. λ π2δ2 Z p d 1 + y 02 dx ≈ (1 + 2 )d = const 0 λ λ E R4 i.e., δ ∝ λ. Thus Ebending ∝ λ2 , Eelastic ∝ α λ2 and Ecompression is not dependent E R4 on buckling wavelength. Minimization of Ebending + Eelastic = λ2 + α λ2 leads to q E λ∝ 4 α R. This scaling result suggests that the buckling wavelength has a weaker dependence on E and α as compared to the dependence on R. 4.3 The mechanical buckling model We propose an elastic buckling model based on the equation of motion of buckling MT bundles to illustrate the buckling phenomenon. We model MT bundles as nested elastic rods (see Fig. 4.1), which buckle against an elastic surrounding network. Let us rst consider buckling of a single rod and then compare it with the buckling of 47 multiple rods. It is known that buckling of a single rod against a xed point by an applied force along its long axis will give rise to Euler buckling, i.e., buckling with half a wavelength. In the case of MT bundles, however, they are not separated but inter- correlated with a dense MT network in the surrounding area between the bundles. Euler type of buckling requires much distortion of the network, thereby increasing the elastic energy of the network. On the other hand, buckling with several bends will reduce the amount of distortion and stabilize the system. To describe the eect of the surrounding medium on buckling, we consider a single bundle in the center of the sample and characterize its interaction with the network using a single elastic constant, α, such that αξ(x) is the elastic restoring force exerted by the network on the bundle per unit length (see Fig. 4.1), where ξ(x) is the bundle displacement in the y direction. Treating the bundle as a rod with a bending rigidity, K, under a compressional force, F, the force balance in the y direction at the onset of the buckling is given by [15, 38, 52]. ∂ 4 ξ(x) ∂ ∂ξ(x) K 4 + [F ] + αξ(x) = 0 (4.1) ∂x ∂x ∂x Standard normal mode stability analysis of Eq. (4.1), assuming ξ(x) ∝ eikx and a uniform F, yields F = α/k 2 + Kk 2 . This result suggests a critical compressional √ force Fc = 2 Kα for a buckling solution and a characteristic wavelength p p 4 λc = 2π/k = π 8K/Fc = 2π K/α (4.2) 48 The resultant characteristic wavelength [Eq. (4.2)] agrees with the prediction for λc based on energy minimization. This model predicts buckling in a higher mode than the fundamental one as in the classic Euler buckling. In agreement with experiments, this model implies that the orientation of MT bundles in a striped sample varies continuously in space in a wave-like manner. In contrast, previous models had suggested that discrete and alternate angular orienta- tions of the MTs formed the striated patterns [89]. In addition, the weak dependence of λc on K and α is consistent with the small variations in both the observed wave- length across a single macroscopic sample and the patterns formed under dierent conditions (for example, samples with dierent tubulin concentrations and samples in containers with dierent sizes). Once the wavelength is selected at the onset of buck- ling, it does not change with time, in agreement with the experimental observation (see Fig. 3.17d). Two limits exist for elastic properties of the bundle (K ). If tight packing (solid model) of the MTs inside the bundle is assumed, then Ksolid = n2 KMT , where KMT ≈ 3.4 × 10−23 N · m2 is the bending rigidity of a single MT [27, 96]. If MTs slide freely inside the bundle, then Kslip = nKMT . By plotting the wavelength over a reasonable range of individual MT lengths for both models[42], we found that the solid model for K appears more reasonable than the slip model, in agreement with previous assumptions, which yield a wavelength comparable to experimental measurements. The fact that K depends quadratically on n in our system suggests that MTs are 49 fully coupled (acting like a solid material) inside the bundle, similar to the behavior of F-actin bundles held together through depletion forces [22]. 4.3.1 Buckling force The occurrence of buckling implies that a longitudinal compressional stress builds up within the MT bundles. The concomitant elongation of the bundles during buck- ling suggests that this stress most likely results from polymerization forces exerted by individual MTs. Other known force generation mechanisms in MT systems re- quire molecular motors, which are absent from these solutions. Thus, we are led to the picture that the individual MTs in a bundle continue to polymerize and, once encountering boundary connements imposed by adjacent bundles and/or cuvette walls, generate a longitudinal compressional force within the bundle. The compres- sional force induced by polymerization increases while MTs keep growing. When the force exceeds a critical value, the bundles start to buckle. In addition this force keeps constant during the process of buckling, consistent with the fact that the elongation rate of MT bundles during buckling is constant (see 3.17f, g and h). 4.4 Numerical estimates about buckling wavelength p 4 Recall from previous discussions that the critical buckling wavelength λc = 2π K/α, where K is the bending stiness of MT bundles and α is the elastic constant of the 50 F y d z 2R Figure 4.2: A schematic showing an array of bundles embedded in an elastic medium. The blue circles represent the cross sections of representative bundles with a spacing of d. The diameter of each bundle is 2R. F indicates the force exerted on the bundles by the surrounding elastic network, which is assumed to be linear with the extent of buckling ξ as: F = αξ dx. Here α is the elastic constant of the surrounding medium and dx represents unit bundle length. The stress σ is: σ = 2RFdx , and strain γ can be written as: γ = dξ . Therefore, shear modulus G0 can be calculated as √ G0 = ξ2R Fd dx = αξ dx d ξ2R dx d = 2R α. So α is related to G0 as α = 2R d G0 = η G0 , where η is the volume fraction of bundles. surrounding MT network. In order to obtain a numerical estimate of the wavelength, we need to estimate K and α. The relation between the bending rigidity of a single MT (k ) and the Young's modulus of single MT (E ) is: k = EI π = E r4 4 So, 51 k E = I 4k = πr4 4 × 3.4 × 10−23 = 3.14 × (12 × 10−9 )4 ≈ 2 × 109 P a Here r is the radius of a single MT. We assume that the Young's modulus of a MT bundle is about the same order of magnitude as that of a single MT (solid model). The spring constant of the network per unit length, α, is proportional to √ the elastic shear modulus of the network in the following way: α∼ 2R d G0 ≈ η G0 ≈ √ 1.36 × 10−3 ×10 P a ≈ 0.4 P a (see Fig. 4.2), where R is the radius of the bundle, d is the lateral spacing between adjacent bundles, G0 is the elastic shear modulus, which is typically in the order of 10 Pa and η is the volume fraction of bundled MTs, which is about 1.36×10−3 3 for 5 mg/ml MT solutions (MT protein density is about 1.4 g/cm ). 5×10−3 g/(1.4 g/cm3 ) So the volume fraction for 5 mg/ml microtubule is: η = 1 ml = 0.36%. Assuming that bundles represent 38% (see Eq. 3.2) of total protein volume, the volume fraction of bundles is 0.36 × 38% ≈ 0.136%). Based on the assumption that MTs are close to tight packing inside a bundle (solid model), the radius of p a bundle is about 250 × (12 nm)2 ≈ 200 nm where 250 is the number of MTs in a bundle and 12 nm is the radius of a single MT. Therefore, an estimation for p √ q q the characteristic wavelength is: λc = 2π 4 K/α = 2 π 5/4 4 Eα R ≈ 6 × 4 10.4GPP aa × 200 nm ≈ 300 µm, which is comparable to the experimentally observed buckling 52 wavelength (see Fig. 3.14A). Part II Optical Tweezers and Its Applications in Microrheology and Caulobacter Motility and Adhesion 53 Chapter 5 Introduction to Optical Tweezers and Microrheology 5.1 How optical tweezers work It was discovered that light intensity gradient in a tightly focused laser beam could be used to trap particles by Arthur Ashkin of Bell Laboratory in 1978 [8]. These traps, also called optical or laser tweezers, allow one to manipulate micron-sized particles with light easily and reliably. They also allow one to take force measurements in the pico-newton range [4, 24, 20]. The understanding of the physical reason for trapping due to a light gradient depends on the wavelength of light used compared to the size of the object trapped. In cases where the dimensions of the particle are greater than the laser wavelength, 54 55 ] 0ÿ 0 ) [ Figure 5.1: Schematic drawing to illustrate origin of optical force in the regime where ray optics can be used. The initial momentum of the light beam is denoted as M and the nal momentum to each photon is denoted M' in the gure. The momentum change is represented by the red arrow. The force thus induced on the sphere points in the opposite direction, as represented by F. a simple ray optics treatment is adequate. As shown in Fig. 5.1, a beam of light is bent by a dielectric sphere. The momentum change of the light beam is shown as the red arrow. Following Newton's third law, the object will gain momentum in the opposite direction [9]. The force induced on the particle due to bending of the beam is shown as F in Fig. 5.1. The net force on the particle is the vector sum of the forces induced by all the light beams of the incident light. 56 On the other hand, if the wavelength of light exceeds the particle dimensions, the particle can be treated as a point dipole in an inhomogenous electromagnetic eld. The force applied on a single charge in an electromagnetic eld is: F1 = q(E1 + dx dt 1 × B), where q is the charge of the particle, x1 is the position of the particle, E1 and B are the electric and magnetic elds at the particle position, respectively. The moment of a dipole is: P = q(x1 − x2 ). Taking into account that the two charges d(x1 −x2 ) have opposite signs, the force can be written as: F = q(E1 − E2 + dt × B) = dP dE (p.5)E + dt × B = α[(E.5)E + dt × B] = α[ 12 5 (E 2 ) + d dt (E × B)] ∼ 0.5α∇(E 2 ), where α denotes the electric susceptibility of the dipole. The second term in the last equality is the time derivative of a quantity that is related through a multiplicative constant to the Poynting vector, which describes the power per unit area passing through a surface. Since the total laser intensity is constant when sampling over frequencies much lower than the frequency of the laser light, which is on the order of 1013 Hz, the derivative of this term averages to zero. The square of the magnitude of the electric eld is equal to the intensity of the beam. Therefore, the result indicates that the force on the dielectric particle, when treated as a point dipole, is proportional to the gradient of the intensity of the beam. The narrowest point of the focused beam, known as the beam waist, is where the electric eld gradient is the greatest. It turns out that a dielectric particle is attracted along the gradient to the region of the strongest electric eld. By directing a monochromatic laser beam through a convex lens it is possible to create a light 57 gradient with the ability to trap a dielectric particle. 5.2 Microrheology Microrheology refers to studies on the elastic properties of polymer networks on mi- crometer scales using microscopic probes. The ability to study them with probes spanning some of the microscopic length scales that are characteristic of polymer net- works (e.g., approaching the inter-chain separation or mesh size of gels) will also lead to important new insights into the microscopic basis of the macroscopic viscoelasticity of such systems. Microrheology readily allows for measuring viscoelasticity at higher frequencies, above 1 kHz or even up to MHz, because the inertia of both the probe and its embedding medium can be neglected at such small length scales [54, 76]. De- pending on whether the probe particle is subject to a forced motion, microrheology can be dierentiated into passive and active ones, which are introduced separately below. 5.2.1 Passive microrheology Passive microrheology uses microscopic particles embedded in the sample and de- tects thermal uctuations of those particles. More specically, a single probe bead is observed at a time by laser interferometry in a light microscope [82]. Detection by photodiodes ensures a bandwidth of detection between about 0.1 Hz and 100 kHz [76]. Frequency-dependent shear elastic and loss moduli are determined from the 58 uctuations of the embedded probes. The response function of the bead is calculated by the uctuation-dissipation the- orem. The procedure is outlined below [82]: The complex single-particle response 0 00 function α(f ) = α (f ) + iα (f ) relates the Fourier transform of the bead displace- ment to the Fourier transform of the force acting on the bead α(f ) = x(f )/F (f ). The uctuation-dissipation theorem provides the link between the Power Spectral Density R t/2 0 0 0 (PSD), S(f ) = limt→∞ 2t xt (f )xt (f )∗ , where xt (f )= −t/2 x(t )e2πif (t ) dt , and the imag- 00 π inary part of the response function by α (f ) = 2kB T f S(f ), where kB is the Boltzmann constant, and T is the solution temperature. A Kramers-Krönig relation can then be R∞ f 0 α00 (f 0 ) used to calculate the real part of the response function [82], α0 (f) = 2 π 0 f 0 2 −f 2 df 0 . Finally, the complex shear modulus is calculated using the generalized Stokes rela- 1 0 00 0 00 tion, G(f ) = 6πaα(f ) , where G(f ) = G (f ) + iG (f ), G (f ) and G (f ) are the elastic and loss moduli, respectively, and a is the radius of the bead. For a spherical probe particle in a purely viscous medium of viscosity η, this expression reduces to the familiar result of G(f ) = i2πf η . The following is a brief derivation of the Kramers-Krönig relation adapted from Toll et al. [93]. The response function α(t) (dened as displacement/force) must be zero for t < 0 since a system can not respond to a force before it is applied. Therefore, the Fourier transform α(f ) is analytic in the upper half complex plane, where f is frequency. Additionally, the response of the probe particle at high frequency (innity) vanishes because there is no time for it to respond to the force before the force 59 has switched direction. We apply the residue theorem for complex integration to α(f ), where both α and f are complex and f resides on the upper half plane. The H α(f ) integration goes as: f −f0 df. The contour encloses the upper half plane at innity, the real axis with the exception a singular pole at f = f0 , and a hump over the pole, leaving no poles inside, so that the integral vanishes. We decompose the integral into its contributions along each of these three contour segments. The segment at innity vanishes since α vanishes as f goes to innity. We are left with the segment along the real axis, and the half-circle, which yields −iπα(f0 ), as predicted by the R∞ α(f ) residue theorem. The sum of these two terms is zero, i.e. −∞ f −f0 df − iπα(f0 ) = 0. Rearranging, we arrive at the compact form of the KramersKronig relation: α(f0 ) = 1 R∞ α(f ) iπ −∞ f −f0 df. Finally, by splitting α(f ) and the equation into real and imaginary R∞ α00 (f ) R∞ α0 (f ) parts, we obtain: α0 (f0 ) = 1 π −∞ f −f0 df and α00 (f0 ) = − π1 −∞ f −f0 df . The KramersKronig relation implies that observing the dissipative response of a system is sucient to determine its in-phase (reactive) response, and vice versa. For reconstructing physical responses, we note that in most systems, the positive frequency-response determines the negative-frequency response because α(f ) is the Fourier transform of a real quantity, so α(−f ) = α∗ (f ). This means α0 is an even func- tion of frequency and α00 is odd. Using these properties, we can collapse the integration range so that it is from 0 to innity. Consequently, we can transform the integral for α0 (f ) into one of denite parity by multiplying the numerator and denominator of R∞ f α00 (f ) f0 R∞ α00 (f ) the integrand by f + f0 and separating as α0 (f0 ) = 1 π −∞ f 2 −f02 df + π −∞ f 2 −f02 df 60 (i.e., the integrand is separated into an even and an odd function). Since α00 (f ) is R∞ f α00 (f ) odd, the second integral vanishes, and we are left with α0 (f0 ) = 2 π 0 f 2 −f02 df . The R∞ α0 (f ) same derivation for the imaginary part yields: α00 (f0 ) = − 2fπ0 0 f 2 −f02 df . 5.2.2 Active microrheology Active microrheology uses microscopic particles embedded in the sample and detects their response to applied forces. The active method involves applying an oscillatory force to a single micron-sized bead. The amplitude and phase of the position of the bead and the applied force are determined, and the complex shear modulus is obtained from the bead response. The equation of motion for a particle driven by an oscillating trap in a viscous medium is determined by two forces: the viscous drag force experienced by the parti- cle, and the force imparted by the optical trap. Hookes' law, with an eective spring constant κ, can approximate the force produced by the laser on the trapped parti- cle. Fig. 5.2 shows the forces on a particle of radius a, in an oscillating optical trap residing in a simple viscous liquid with viscosity η0 . For a simple viscous liquid, the drag force is taken to be the Stokes drag Fdrag = −6πη0 av , where η0 is the zero shear viscosity of the liquid, and v is the velocity of the particle. For a micron-sized particle, this expression for the viscous drag is an approximation because the inertia of the uid has been neglected (low Reynolds number ow), making the expression valid for relatively low oscillation frequencies. 61 )GUDJ )WUDS [ ZW $HLZW Figure 5.2: Schematic showing a damped driven harmonic oscillator, which illustrates the motion and the forces on a particle using an applied oscillatory laser trap. The solid line denotes the origin of the coordinate system, the dashed line indicating the location of the bead, and the dotted line the center of the oscillating trap. The equa- x(w, t) = −6πη(w)ax(w, tion of motion for the particle is: m¨ ˙ t) − κ(w)x + kot Aeiwt − kot x(w, t), where kot is the trap stiness of the oscillating beam, A is the amplitude of the oscillatory beam with frequency w, η(w) and κ(w) represent the viscosity and the elasticity of the medium, respectively, and x(w, t) is the bead displacement with respect to the center of the trap (solid line). 62 The force due to the trap is given simply as the eective spring constant multiplied by the distance between the center of the particle and the center of the trap. New- ton's second law of motion then leads to the equation of motion for the particle as: x(w, t) = −6πη(w)ax(w, m¨ ˙ t) − κ(w)x + kot Aeiwt − kot x(w, t), where kot is the trap stiness of the oscillating beam, A is the amplitude of the oscillatory beam with fre- quency w, η(w) and κ(w) represent the viscosity and the elasticity of the medium, respectively. Here x(w, t) is the time-dependent position of the particle as a function of oscillation frequency: x(w, t) = x(w)eiwt = D(w)ei(wt−δ(w)) (5.1) To relate the motion of a single particle to the mechanical properties of the mate- rial surrounding the particle, the complex response function α(w), which is the ratio of the displacement of a particle to the external forces on the particle is calculated. The 0 00 eective storage modulus, G (w), and eective loss modulus, G (w), are then obtained x(w) D(w)e−iδ(w) from 1 k = ot 6πaα(w) 6πa + G(w) = kot 6πa + G0 (w) + iG00 (w), where α(w) = F (w) = Akot . Therefore, the storage and loss moduli of the medium can be calculated from the measured phase shift and amplitude of the particle motion using the following rela- tionships: 0 κ(w) kot Acos(δ(w)) G (w) = = ( − 1) (5.2) 6πa 6πa D(w) 00 kot Asin(δ(w)) G (w) = wη(w) = ( ) (5.3) 6πa D(w) Chapter 6 Setup of the Optical Trapping and Position Detection System 6.1 Optical tweezers setup The optical tweezers set up is illustrated in Fig. 6.1. The laser used is Diode Pumped (CrystaLaser) and has a maximum power of 500 mW. The 1064 nm wavelength is within a window of spectrum of minimal water absorption and cell damage. The laser beam rst goes through an optical isolator (Thorlabs, Inc.), which prevents the reected beam from re-entering the laser. Then the beam passes through a beam expander (CVI Laser, LLC), which lowers the incident power per area to reduce the wear on the optical components. A power attenuator made by combining a Glan Laser Prism (Zeta International) and a half-wave plate (CVI Laser, LLC) can 63 64 Lens Quardrant Dichroic Photo Diode Mirror Filter Amplifier Data Acquisition Condenser Board, Computer Sample Objective Dichroic Beam Raiser Mirror IR Filter Glan Half Laser Wave Beam Beam Optical Nd:YAG Laser Prism Plate Expander Expander Isolator λ = 1064 nm Nikon TE2000 Beam Dump Microscope Beam Dump Figure 6.1: Laser tweezers setup. adjust the laser intensity. After the power attenuator, the beam goes through the second beam expander (Rodenstock) to increase the beam size enough to ll the back of the objective. The beam raiser consists of a stable rod and two mirrors set at 45°. After entering the inverted microscope (TE2000U, Nikon), the beam is reected by a 1064 nm dichroic mirror (Part No. 900dcsp, Chroma Technology). The dichroic mirror reects the laser beam upwards through the microscope objective while allowing visible light from the microscope light source above to pass downwards, and then reected by another mirror to a FastCam PCI camera (FASTCAM-PCI R2, Photron). An IR lter is placed below the dichroic mirror to reduce the amount of residual laser light reected by the sample from reaching the camera. An upper dichroic mirror of the same specication mounted at 45o relative to the light path is used to direct the laser light from the back focal plane of the condenser to a lens. Finally, the laser beam reaches the quadrant photodiode detector (QPD). Room light is removed by a lter in front of the detector. 65 6.2 Position detection using photodiode detector A Gaussian laser beam is brought to a focus at the sample by an objective lens and leads to the trapping of dielectric spheres as described previously. The condenser lens collects the diverging beam from the sample and images the laser focus where the bead is trapped. The trapped bead acts like a point-dipole scatterer. The incident light and the scattered light o the bead interfere and produce an intensity pattern on the back focal plane (BFP) of the condenser. The intensity distribution on the BFP of the condenser is then detected by a quadrant photodiode. The signals from the four quadrants of the photodiode are combined and dier- entiated to obtain X and Y signals corresponding to the displacements of the bead in these directions in the plane normal to the propagation direction of the laser. For ex- ample, the x-position can be calculated by obtaining the total intensity and intensity I+ −I− 16kα x −x2 /w2 dierence in the two halves of the diode as follows: I+ +I− = √ πw2 w e , where k= 2πns λ and λ is the wavelength of light, ns is the refractive index of light in the solvent, w is the radius of the eld on the focal plane and α is the electric susceptibility. As I+ −I− x << w, I+ +I− ∝ x. The output signals are, after analog amplication and pre-processing by the dif- ferential amplier, acquired as voltages using a BNC 2090 board and digitized using software written in Labview (National Instruments, Austin, TX). 66 6.3 Calibration of the optical trap 6.3.1 Theory Various methods have been used to measure trap force in the linear region [70, 102, 31]. The method outlined below is commonly used to obtain trap stiness. The equation of motion of a bead trapped in buer solution is given by the equation: β dx dt +κx = F (f ), where β is the hydrodynamic drag coecient and κ is the trap stiness. F (f ) is the external random force due to Brownian Motion. The power spectral density (PSD), R t/2 0 0 0 S(f ) = limt→∞ 2t xt (f )xt (f )∗ , where xt (f )= −t/2 x(t )e2πif (t ) dt , can be shown to have kB T α a Lorentzian form: Sxx (f ) = π 2 β(f 2 +fc2 ) , which has a roll-o frequency fc = 2πβ , from which the trap stiness can be calculated as: α = 2πβ fc . A position calibration factor for each laser power relates the quadrant diode output voltage to the displacement of the trapped bead with respect to the trap center. The position calibration factor is obtained by observing the thermal uctuations of a bead in a buer solution taken at the same power settings as for the actual samples of interest. Then a power spectral density Svv (f ) was calculated from the voltage Svv (0) fc2 time series. We t Svv (f ) to a Lorentzian function Svv (f ) = (f 2 +fc2 ) , with Svv (0) and fc as the tting parameters. Svv (f ) is related to the position power spectrum Sxx (f ) by Svv (f )= ρ2 ·Sxx (f ) (this equation can be taken as the denition for the conversion factor), where ρ represents the position conversion factor (in volt/nm). Take f =0 kB T Hz, we obtain Svv (0) = ρ2 Sxx (0). Notice that Sxx (0) = π 2 β fc2 , assuming that the 67 Figure 6.2: Scattering force, gradient force and total trap force on a one micron spherical particle as a function of the displacement of the particle to the trap center. This result is derived by A. Ashkin [9]. 68 motion of the bead in the buer is that of Brownian motion in pure water. So, 2 2 Svv (0) = ρ2 πk2Bβ Tf 2 . Solving for ρ, we get ρ=( Svv (0) fc π β 0.5 kB T ) , where β is the viscous drag c coecient of the particle in water and Svv (0) and fc are obtained tting parameters mentioned earlier. Most optical traps are operated in such a way that the dielectric particle rarely moves far from the trap center. The reason for this is that the force applied to the particle is linear with respect to its displacement from the center of the trap as long as the displacement is small. In case of an application out of the trap center region, however, one needs to know the whole range force prole. Over the whole range of the trap, Ashkin derived theoretically the total optical force on a bead as a function of distance between the focus of the laser beam and the center of the bead using geometric optics treatment [9]. Gittes et al. developed a method of calibrating the whole range trap force based on the fact that the rate of light momentum transfer is equal to the lateral trap force [36, 6]. Their experimental results were obtained using bead diameter smaller than the laser wavelength and the model they developed was based on small-particle scattering. 6.3.2 Experimental determination of trap stiness and whole range trap force We probed the force prole both in the linear region of an optical trap and in the region where the trap force decreases gradually. Our measured force prole using 69 particle size of about 1.5 µm and laser wavelength of 1064 nm appears similar to that of the total optical force derived theoretically by Ashkin (Fig. 7 in [9]). Although the conditions of our measurements and the assumptions of the Ashkin derivation are not exactly the same, the force prole we obtained provides the actual strength and working range of the optical trap. The trap stiness near the center of the trap was rst obtained from the power spectral density (PSD) of a trapped silicon bead of 1.5 µm diameter in water. The position uctuation of the trapped bead was detected by a photodiode detector. The power spectral density of the bead was calculated and t to a Lorentzian function kB T α [87, 3], Sxx (f ) = π 2 β(f 2 +fc2 ) , which has a roll-o frequency fc = 2πβ , where α is the trap stiness and β is the viscous drag coecient. Fig. 6.3a shows the data and t of the power spectral density of a typical bead. From the value of the tting parameter fc (∼ 40 Hz), we calculated the trap stiness α = 2πβfc ≈ 0.0035 pN/nm. The trap force near the center of the trap can also be obtained from the trap potential, assuming Boltzmann distribution of displacements of trapped particles. The probability distribution of displacement x of a typical bead p(x) at the same laser intensity was used to obtain the trap potential V (x) based on V (x) = −kB T ln(p(x))+ const. The potential curve V (x) was dierentiated with respect to x to obtain the trap force. Fig. 6.3b shows the trap potential and a parabolic t (dashed curve). Fig. 6.3c shows the corresponding trap force values (gray dots) and a linear t (dashed line). Shown in comparison is a solid line with a slope equal to the trap stiness 70 2 a 10 PSD (nm 2 /Hz) 0 10 −2 f -2 10 −4 10 −2 0 2 4 10 10 10 10 frequency (Hz) 8.5 b 8 trap potential (K B T) 7.5 7 6.5 6 5.5 −50 0 50 0.3 c 0.2 trap force (pN) 0.1 0 −0.1 −0.2 −50 0 50 displacement (nm) Figure 6.3: Measurement of trap force in the linear region of the trap. (a) Power k T spectral density of a typical bead and a Lorentzian t Sxx (f ) = 2 B2 . (b) Trap π β(f +fc2 ) potential as a function of displacement and a parabolic t. The potential is calculated from the distribution of bead displacement at the same laser intensity as that used in (a). (c) Comparison of trap force obtained from (a), denoted by the solid line, and (b), denoted by the gray dots and a linear t to them shown as the dashed line. 71 Figure 6.4: A picture of the ow cell. The cell is about 15 cm long and 5 cm wide. At the end of the cell are inlet and outlet tubings. The two outlets in the middle are connected to vacuum. A gasket is attached to the bottom of the cell (green) to ensure good isolation from the outside air and prevent leaking of solution. calculated from Fig. 6.3a. The slopes of the two lines agree within 14 %. As can be seen from the linear t to the gray dots in Fig. 6.3c, the relationship between the force and the displacement is approximately linear. Because the trapped bead rarely explored displacements that were beyond 100 nm from the trap center, experimental measurements of the trap force in the 100-500 nm range could not be obtained via the Boltzmann distribution method due to poor statistics. In summary, the agreement of the trap force obtained by the power spectral den- sity and the Boltzmann distribution analysis conrms the validity of both methods. The power spectral density method yields reliable result for the trap stiness. The Boltzmann distribution method yields more directly the trap potential and the corre- sponding force. A drawback of the Boltzmann distribution method, however, is that the probability for the trapped particle to reach displacements far from trap center 72 a 90 80 bead velocity (µm/s) 70 60 50 40 30 0 0.5 1 1.5 2 b 0.8 0.6 force (pN) 0.4 0.2 0 0 0.5 1 1.5 2 displacement (µm) Figure 6.5: (a) The velocity of a typical bead initially trapped but then dislodged upon application of ow in the ow chamber. (b) The solid circles represent the force on the same bead as in (a) and were t to an exponential function to yield y = 1.4 × e−1.5 x (dashed curve). In the linear region, the gray dots (red) within a displacement of about 100 nm are replots from Fig. 6.3c, and the dashed line depicts the expected force prole in this region. 73 becomes exceedingly small. Therefore, the statistics becomes inadequate to determine the potential reliably at large displacement. Nevertheless, the force it determines in the low displacement region is reliable. We measured the trap force at the far edge using micron-sized beads in a ow chamber. A ow cell (GlycoTech) and a 170 µm thick glass coverslip formed the sample chamber. Flow was injected using a syringe lled with 1.5 µm diameter beads in water. The laser beam was focused inside the sample chamber and a bead was trapped before ow was applied. After the application of ow, the bead started to dislodge from the trap center when the ow speed increased and reached a threshold value. Several trials were performed in advance in order to optimize the progressing speed of the syringe pump, which produced the threshold ow speed. The trajectory of the escaping bead was recorded by the fast camera at 1000 fps, and the speed of the bead was measured along the trajectory. The trapping force was approximated as the following: ftrap = 6πηa(vf low − v bead ), where the radius of the bead a is 0.75 µm, and vf low and v bead are the velocities of the ow and the bead, respectively. Here vf low was taken as the terminal velocity of the bead. The speed of a typical bead in the ow chamber was plotted in Fig. 6.5a. The calculated force was plotted in Fig. 6.5b, denoted by solid circles, which were t to an exponential function y = 1.4 × e−1.5 x (dashed curve). Fig. 6.6 shows a schematic of the trap and the whole range trap force prole. The numbers represent displacements in micro-meters, with 0 indicating the center of the 74 WUDSIRUFH S1  GLVSODFHPHQW µm Figure 6.6: An illustration showing the force prole and range of a laser trap. A picture of the trap is shown in red, a probe particle in blue, and the whole force prole in black. The dashed line indicates the position of the beam waist. The dimensions are drawn roughly to scale. The trap force is linear with displacement in the low displacement regime, consistent with experiments. The force drops gradually down to zero beyond the peak when the bead is so far away from the trap center that no laser beams are scattered by the bead. trap. The picture is drawn roughly to scale. An image of the probe particle is also shown. Chapter 7 Application of Optical Tweezers in Microrheology of Microtubule and Actin Networks 7.1 Experimental methods 7.1.1 Measurement of viscoelasticity of isotropic and nematic actin networks using passive microrheology Actin is a dominant cytoskeletal protein present in almost all eukaryotic cells. It plays an essential role in a variety of cell functions, such as motility, shape change, and mechano-protection [10]. For protein concentration up to a few mg/ml, actin 75 76 laments form an entangled isotropic network. Above a threshold concentration, F- actin solution undergoes an isotropic to nematic liquid crystalline phase transition [23, 32, 100]. Quantitative measurement of the elastic properties of actin networks in both phases is an essential rst step in understanding the elasticity of cytoskeletal networks and the pivotal role of actin binding proteins in the mechanical behavior of the cell. Actin was extracted from rabbit skeletal muscle following the technique of Pardee and Spudich [75] and kept in G-buer (2 mM Tris HCl, pH 8.0, 0.2 mM ATP, 0.2 mM CaCl2 , 0.2 mM DTT, and 0.005% NaN3 ) at −80 ◦ C (G-actin). For experiments, ◦ aliquots of G-actin was thawed rapidly to 25 C and centrifuged for 5 min at 7000 g. The G-actin was then polymerized to form F-actin by adding KCl and MgCl2 to nal concentrations of 50 mM and 2 mM, respectively. For our experiments, the average lament length of F-actin was measured to be approximately 7 µm. The beads used in our experiments are silica beads with a diameter of 1µm (Bangs Laboratories Inc., Fisher, IN). Bead suspensions were pre-diluted with the same buer solution as that of F-actin before they were added to F-actin solutions in order to enable proper observation of beads motion without altering the ionic condition of F- actin. The mixed solutions were injected into rectangular capillary tubes (VitroCom 2 Inc., Mt. Lakes, NJ) with cross-sectional dimensions of 0.1 × 1 mm . The tube long axis was dened as the x direction, and the y direction was perpendicular to the x direction and in the plane of observation. Due to the shear alignment when the 77 sample was lled, the nematic director was along the x direction for F-actin in the nematic phase. Each capillary was immediately sealed with an inert glue to eliminate ow and evaporation from the system. For passive microrheology measurements, we use a Nikon Eclipse TE 2000 inverted microscope. After loading a sample, the laser beam was focused at a sucient height (larger than 10 µm) above the bottom surface of the sample chamber to minimize boundary eect. Displacement data for each bead was recorded until the desired number of data points were acquired (usually 540,000 or 1,080,000 data points per channel). Power spectral densities and Kramers-Krönig integrals were calculated using software written in C++. The complex shear modulus is calculated using the 1 0 00 generalized Stokes relation, G(w) = 6πaα(w) , where G(w) = G (w)+iG (w), as follows: 1 α0 (ω) G0 (ω) = , (7.1) 6πa [α02 (ω) + α002 (ω)] 1 −α00 (ω) G00 (ω) = . (7.2) 6πa [α02 (ω) + α002 (ω)] For nematic solutions, αx (ω) and αy (ω) are dierent. Assuming decoupling of the motion in x and y directions, we consider separate response functions, x(ω) = αx (ω)Fx,ω and y(ω) = αy (ω)Fy,w . Using Eqs. 7.1 and 7.2, we obtain G0x (ω), G00x (ω), G0y (ω) and G00y (ω). The G00 (ω) that is measured experimentally incorporates the viscous modulus of both the buer and the polymer network of interest. The solvent contribution 2πηf was subtracted from G00 (ω), where η is the viscosity of the buer solution (taken 78 H a lo g e n Lam p O p tic a l F ilte r P o s itio n S e n s in g D ic h ro m a tic M irro r D e te c to r S a m p le CDSR L o c k - in A m p lifie r OBJ M ic ro s c o p e CCD PC T e le s c o p e F u n c tio n G e n e ra to r B e a m C o m b in e r 633 nm 1064 nm Beam Expander P Z T - d riv e n m irro r Figure 7.1: A schematic diagram of the active microrheology experimental setup [104]. The area enclosed by the dashed lines represents an inverted optical microscope. approximately as that of water). Since the probe particle experiences an elastic restoring force from the trap in addition to the elastic force from the medium, there is κ(f ) an elasticity contribution from the trap. Based on G0 (f ) = 6πa , where κ(f ) represents the elasticity of the medium, the contribution to elasticity due to the trap stiness is kot /(6πa), where kot is the trap stiness and a is the bead radius. This contribution has been subtracted from the measured G0 values at every frequency. 79 7.1.2 Measurement of anisotropic elastic properties of MT bundled network using active microrheology The use of oscillatory optical tweezers to trap and oscillate a particle embedded in an elastic medium and to measure its mechanical response has been demonstrated elsewhere [48, 98]. Fig. 7.1 shows the schematic diagram of the setup of an oscillat- ing optical tweezer-based microrheometer used at Professor OuYang's lab at Lehigh University, where the active rheology experiments were conducted. The expanded beam has a size slightly larger than the back aperture of the microscope objective OBJ to achieve an  overll, or full numerical aperture condition. An infrared laser (Nd:YVO4 1064nm diode-pumped solid-state cw laser, Spectra Physics) was steered by a mirror mounted on a PZT (piezoelectric transducer, which converts electrical pulses to mechanical vibrations) 2-axis platform (S-330, Physik Instrument) which was connected to a function generator (a built-in function of the lock-in amplier, Stanford Research SR-830) [104]. The active element of PZT is basically a piece of polarized material (i.e. some parts of the molecule are positively charged, whereas other parts of the molecule are negatively charged) with electrodes attached to two of its opposite faces. When an electric eld is applied across the material, the po- larized molecules will align themselves with the electric eld, resulting in induced dipoles within the molecular or crystal structure of the material. This alignment of molecules will cause the material to change dimensions. The 2-axis platform allows the trapping beam to oscillate in two orthogonal directions (x and y), thus enabling 80 the study of mechanical properties in two orientations. The 1064 nm and 633 nm HeNe laser beams (Uniphase, 5mW) joined with a beam combiner cube into a collinear conguration were launched into the right side port of an inverted microscope (Olympus IX-81) via a telescope lens pair, and directed into the direction of microscope optical axis via a dichromatic mirror (not shown). An oil immersion objective lens (OBJ) (NA=1.3, 100X, Olympus) was used to focus both laser beams to a common focus. The 1064 nm beam was used to trap and the 633 nm beam was used to track the position of dielectric probe particles inside the sample. The tracking beam, diracted by the moving probe particle, was collected by the condenser CDSR (NA = 0.9) and projected onto a dual-lateral position-sensing detector (PSM2-4 and OT-301, On-Trak) where the motion of the probe particle in the transverse sample plane (X-Y plane) was tracked in real-time. An optical lter was inserted to prevent 1064 nm beam from reaching the detector. The lock-in amplier, fed with signals from the detector and referenced with the sinusoidal electric signal that drove the PZT-driven mirror, provided the displacement amplitude and phase shift of the trapped particle with great sensitivity. A CCD camera was installed on the microscope left-side-port to record bright-eld images. The dichromatic mirror reected laser beams and passed visible lights from the microscope halogen lamp so that sample imaging and rheological measurements could be performed simultaneously. The lock-in amplier, PZT-driven mirror, and the microscope (including the CCD camera) were controlled by a PC. 81 We applied an oscillatory optical force to a trapped bead and analyzed the os- cillatory motion of the particle to determine the viscoelastic moduli of the media. The trapped bead was forced to oscillate by the oscillatory tweezers driven by the PZT-controlled mirror. The trap spring constant was determined by applying the equations for the am- plitude and phase shift of bead displacement on water. The frequency-dependent amplitude and phase shift data were t to the following equations [98, 104]: q 6πη0 a D(w) = √ A , δ(w) = tan−1 ( 1−wτ 2w/w2 ). Here τ = kot t , w0 = kot m . τ 2 w2 +(1−w2 /w02 ) 0 The spring constant kot thus deduced agreed with that measured by using Boltzmann statistics for a particle in a parabolic potential well. Microtubule samples were prepared as follows: 3.6 mg/ml tubulin with 1.5 µm diameter silicon beads added was polymerized into MTs (taxol.OG was added with a molar amount 3.5% that of tubulin dimers to promote bundle formation and to label MTs). A chamber made of a glass slide and a coverslip with a thickness of about 100 µm was sealed with vacuum grease rst and then further sealed using an inert glue. The slide was aged for about 24 h before measurements in order to minimize drift. In a typical measurement, a bead in the slide was located using bright eld microscopy. MT bundles around the bead can also be examined using uorescence microscopy. Confocal uorescence images were taken, displaying the MT network around the bead. 82 7.2 Results and discussion 7.2.1 Moduli as a function of frequency for isotropic and ne- matic F-actin solutions We use passive microrheology to measure G' and G for isotropic actin solutions. We also characterized anisotropic viscoelasticity of nematic F-actin solutions, with the network more viscoelastic in the direction perpendicular to the nematic director. To our knowledge, this is the rst time that the rheological properties of F-actin in the aligned phase are investigated. There have been many rheological studies on conventional nematic polymer liq- uid crystals [7, 106, 11, 19]. Among the properties investigated are shear modulus as a function of concentration or temperature, and viscosity as a function of shear rate. However, few previous studies cover the anisotropic behavior of shear modu- lus in the nematic phase. Recently, a piezo-rheometer technique has been used to measure the complex shear rigidity modulus as a function of frequency and temper- ature of elastomers and polymers [33, 86, 61]. Martinoty et al [61] measured G' of nematic elastomers in two geometries, with shear rate perpendicular and parallel to the nematic director, respectively. This study shows anisotropy of viscoelasticity of nematic elastomers, with G0y larger than G0x . The lowering of G0x in the nematic phase was attributed to the coupling between the shear and the director. Though the ne- matic elastomer with covalent crosslinks is dierent from the nematic F-actin network 83 without crosslinks, these two systems do have similarly anisotropic viscoelasticity. For both isotropic and nematic actin solutions, the displacements of the probe bead were recorded by the quadrant photodiode as voltage outputs which are pro- portional to displacements. Voltage outputs for two perpendicular directions are acquired by a BNC 2090 board and processed by LabVIEW. Then the response func- tion α∗ (ω) = α0 (ω)+iα00 (ω) was calculated from power spectral density of the particle motion. Typical results for a 0.5 mg/ml isotropic actin solution are shown in Fig. 7.2. Since the power spectral density was calculated from the displacements, which are discrete and cover only a limited time range, of the probe particle via Fast Fourier Transform, signicant truncation errors near the frequency extremes were introduced, whereas it does give accurate results within the frequency region between the two extremes. To gain a physical understanding of the moduli, consider the two distinctive frequency regimes. At low frequencies, the elastic restoring force of the trap dominates the motion of the probe particle, so G' is larger than G. At high frequencies, the viscous damping force dominates the motion of the particle, so G is larger than G'. Note that the moduli for the two orthogonal directions (x and y) overlap with each other, displaying isotropic viscoelasticity. As we have noticed in Fig. 7.2, the data in low frequency range (below 1 Hz) is very noisy. In order to probe the low frequency viscoelastic properties more accurately, lower sampling frequencies (number of data points not changed) were used. Results 84 102 Gx’ 1 10 Gx" Gy’ Gy" PRGXOL 3D 0 10 -1 10 -2 10 -3 10 -2 -1 4 10 10 100 10 1 102 3 10 10 frequency (Hz) Figure 7.2: Moduli for a typical 0.5 mg/ml isotropic F-actin solution (contribution to elasticity due to the trap stiness has been subtracted). Filled symbols represent G' and empty symbols represents G. Black symbols represent data for the direction perpendicular to the sample capillary. Red symbols represent data for the direction parallel to the capillary. The sampling frequency was 20000 Hz and 540000 data points were recorded. with a sampling frequency of 2000 Hz and 540000 data points are shown in Fig. 7.3. By reducing the sampling frequency to 2000 Hz while maintaining the same number of data points (thus increasing the total measurement time), we were able to obtain more reliable low frequency information. For the 1 mg/ml isotropic F-actin solution, G0x and G0y dier very little, and the same statement holds for G00x and G00y , showing isotropic viscoelasticity. The slight dierence could be due to the weak metastable lament alignment caused by the shear ow as the sample was lled into the capillary. The G0 s have a weak dependence on frequency, showing the characteristic of a plateau modulus. The values of G0 agree with those measured by Gardel et al. [34]. The G00 s increase with frequency approximately in a scaling relation of G00 ∼ ω 3/4 , consistent with the microrheology 85 3 10 2 10 1 10 moduli(Pa) 0 10 Gx’ -1 Gx" 10 Gy’ -2 Gy" 10 -3 10 -4 10 -3 3 10 10 -2 10 -1 100 10 1 102 10 frequency (Hz) Figure 7.3: Moduli for a 1 mg/ml actin solution (contribution to elasticity due to the trap stiness has been subtracted). Filled symbols represent G' and empty symbols represents G. Blue symbols represent data for the direction perpendicular to the sam- ple capillary. Green symbols represent data for the direction parallel to the capillary. The sampling frequency was 2000 Hz and 540000 data points were recorded. measurements by Gittes et al. [37], Xu et al. [108], and Mason et al. [63], and also with the recent optical tweezers passive microrheology results by Brau et al. [17]. The result (with a sampling frequency of 20,000 Hz, 540,000 data points) for a 4 mg/ml nematic actin solution is shown in Fig. 7.4 (LDPT). Shown together is the result for 4 mg/ml nematic actin solution obtained using video particle tracking (VPT) method, in which videos of bead motions are recorded, the positions of beads are tracked using MetaMorph software, and the displacements of the probe particles are measured. For the 4 mg/ml nematic F-actin solution, results for G0x , G0y and G00y agree with video particle tracking method. But the result for G00x is one half of that measured by video particle tracking method in the overlapped frequency region. G0x and G0y both 86 2 10 Modulus (Pa) 1 10 0 10 G’ x LDPT VPT G’ y −1 10 10 2 ω3/4 Modulus (Pa) 1 10 0 10 G’’ x LDPT VPT G’’ y −1 10 −1 0 1 2 3 10 10 10 10 10 ω (rad/sec) Figure 7.4: Frequency dependence of moduli for a 4 mg/ml nematic F-actin solution. For the spectra on the left, the bead displacement was measured with the VPT (video particle tracking) method and the mean square displacement was converted to moduli using Generalized Stocks Relation; for the spectrum on the right, the bead positions were tracked by the photodiode. In the overlapped region between the two dash dot lines, the laser tweezers based results are shown in gray symbols. The two methods 0 0 00 00 give consistent results for Gx , Gy , and Gy ; Gx measured by the VPT method, however, is about twice of that measured by laser tweezers based method. Both spectra were measured using 1 µm beads and obtained from averaging over ten beads. The solid 3/4 straight line shows a scaling law of ω . 87 show the behavior of plateau moduli. G00x and G00y follow a scaling behavior of ω 3/4 , consistent with recent optical tweezers passive microrheology results by Brau et al. [17]. The anisotropic viscoelasticity of nematic F-actin solutions can be interpreted based on the tube model [28, 65], which states that the motion of an individual lament in an isotropic network can be described as diusing in a virtual tube con- structed by its neighboring laments (For a bead of diameter 1 µm embedded in an isotropic actin network, the mesh size is ξ = 0.3 c−0.5 = 0.3 µm, where c=1 mg/ml. For a bead with diameter of 1 µm embedded in a nematic actin network, the average spacing of two neighboring laments is estimated to be d = 0.17 c−0.5 = 0.085 µm, where c=4 mg/ml). Consequently, the transverse diusion of the lament is signi- cantly suppressed, whereas its longitudinal diusion is mostly unhindered. Since the transverse diusion of the laments is suppressed, the laments whose contours are tangential to the bead experience more restriction in transverse diusion and thus can hardly vacate room for the bead to diuse, whereas the laments pushing the bead with their ends can more easily diuse away longitudinally, hence leaving more room for the beads to diuse. In other words, the actin network is more resistant to perturbations in the transverse direction. As the complex shear modulus measures the resistance of material to deformations, it is reasonable that the modulus in the transverse direction is higher. In summary, the laser tweezers based measurements are capable of giving high 88 frequency information because the detector responds extremely fast to changes of the bead position. The method readily acquires a large number of data points, which can be eciently processed by direct Fourier transform. The viscoelastic properties are isotropic for isotropic F-actin solutions, whereas nematic phase F-actin solution is more elastic in a direction perpendicular to the nematic director. 7.2.2 Anisotropic elastic properties of bundled MT network MTs were labeled with taxol conjugated Oregon Green and imaged using spinning disc confocal microscopy, which enables us to observe the orientation of bundles during rheology measurements. Spinning disc confocal is capable of high spatial resolution and rapid image acquisition. The spinning disc consists of a thin wafer with hundreds of pinholes that are arranged in a spiral pattern. When a portion of the disc is placed in the internal light path of the confocal microscope, the spinning disc produces a scanning pattern of the subject. As the subject (exactly on the focal plane) is inspected, light is reected back perpendicularly through the microscope objective. However, light that is reected from in front of or behind the focal plane of the objective approaches the disc at an angle rather than perpendicularly. The pinholes of the disc permit only perpendicularly oriented rays of light to penetrate. This enables the microscope to view a very thin optical section of tissue, at extremely high speed because it is not limited by the speed of a scanning laser. A typical superimposed confocal and bright eld image is shown in Fig. 7.5A. 89 Three beads of 1.5 µm diameter are labeled. Bead 1 is embedded in a bundle. The oscillating laser beam is able to oscillate in two orthogonal directions, x and y, re- spectively. The amplitude and phase (with respect to the oscillating beam) of the bead motion were recorded. The storage and loss moduli of the medium were then calculated from the measured phase shift δ(w) and amplitude D(w) of the particle motion. The result of active rheological measurements for bead 1 is shown in Fig. 7.5B. Both moduli fall into the range of 5-500 Pa. Data at high frequency are noisy due to uid inertia. Moduli in the directions along (x direction) and perpendicular (y direction) to the bundle orientation dier as expected. Also as expected, G'>G at low frequencies and G'