Microrheology and Dynamics of F-actin Networks by Jun He M. Sc., Brown University, 2007 M. Sc., Nanjing University, China, 2004 B. Sc., Nanjing University, China, 2001 Thesis Submitted in partial fulfillment 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 “Microrheology and Dynamics of F-actin Networks” by Jun He, Ph.D., Brown University, May, 2009. We study the rheological properties of reconstituted actin networks by tracking the thermal motion of embedded micron-sized probe beads with four types of surface coatings. For the most slippery beads, thermal motion causes those smaller than the network mesh size to percolate through the network or hop from one confinement “cage” to another. Consequently, the smaller beads sense a weaker network. This trend is reversed for three other types of beads, which detect an apparently stiffer network due to the physisorption of nearby filaments to the bead surface. We also confirm the existence of a depletion layer around non- or weakly-sticky probe surfaces by confocal imaging. Analysis of these effects is necessary in order to accurately de- fine the actin network rheology both in vitro and in vivo. We investigate microrheological properties of F-actin across the isotropic-nematic phase transition region by both video particle tracking and laser deflection particle tracking. As the nematic order parameter increases with actin concentration, the storage modulus in the perpendicular direction grows faster and larger than that in the perpendicular direction. Furthermore, we find that the viscoelasticity of F-actin network varies with the magnesium concentration more sensitively in the nematic phase than in the isotropic phase. In all, particle tracking microrheology reveals rich rheological features of F-actin affected by the isotropic-nematic phase transition and by tuning weak electrostatic interactions among the protein filaments. To address the network dynamics, we observe an abnormal slowdown of the lon- gitudinal diffusion of F-actin across the isotropic-nematic phase transition region. We also find that the F-actin diffusion across the transition region markedly differs from the diffusion of microtubule and fd virus in F-actin solutions. Additionally, the viscous drag probed by F-actin is found to increase sharply with magnesium con- centration in the nematic but not in the isotropic state. Based on these results, we propose that the abnormal slowdown is caused by the counterion induced transient association between parallel actin filaments in the nematic phase. In conclusion, a number of physical properties of F-actin have been better de- fined through this thesis work. The new findings shed lights on a range of biological functions of actin based structures, including cell mechanics and motility. © Copyright 2009 by Jun He This dissertation by Jun He is accepted in its present form by the Department of Physics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Jay X. Tang, Director Recommended to the Graduate Council Date Thomas Powers, Reader Date James M. Valles Jr., Reader Approved by the Graduate Council Date Sheila Bonde Dean of the Graduate School iii Vita Education • Brown University, Providence, RI 02912 2004-2009 – Ph.D. in Physics. Department of Physics. May 2009 – M.Sc. in Physics. Department of Physics. May 2007 • Nanjing University, Nanjing, P.R. China 1997-2004 – M.Sc. in Physics. Department of Physics. July 2004 – B.Sc. in Physics. Department of Physics. July 2001 Publications and Manuscripts • Jun He and Jay X. Tang, “Probe Size Dependent Microrheology of F-actin Network Regulated by Surface Chemistry”, (in preparation 2009) iv • Jun He, Michael Mak, Yifeng Liu, and Jay X. Tang, “Effect of Mg Ion on Microrheological Properties of F-actin Solution across Isotropic-Nematic Phase Transition”, Physical Review E, 78, 011908 (2008) • Jun He, Jorge Viamontes, and Jay X. Tang, “Counterion-Induced Abnormal Slowdown of F-Actin Diffusion across the Isotropic-to-Nematic Phase Transi- tion”, Physical Review Letters, 99, 068103 (2007) • Xuenong Ying, Jun He, and Yening Wang, “A low temperature internal fric- tion study of Y0.85 Ca0.15 Ba2 Cu3 O7−δ with different oxygen content”, Solid State Communications, 130, 441 (2004) Academic Experience • 6/2005 - 5/2009: Research Assistant. Biophysics Lab, Department of Physics, Brown University, Providence, RI. • 9/2004 - 5/2005: Teaching Assistant. Department of Physics, Brown Univer- sity, Providence, RI. Awards and Honors • Coline M. Makepeace dissertation fellowship, Brown University, 2008-2009 v • The People’s Scholarship for Outstanding Students, the 1st grade, Nanjing Uni- versity. 1998-2000 Professional Society • American Physical Society, member since March 2007 • Biophysical Society, member since February 2009 vi Preface This thesis provides a microscopic investigation of mechanical properties and dy- namics of filamentous actin (F-actin) networks via micron-sized particle tracking mi- crorheology and single filament tracking technique, respectively. This work focuses on how these properties or behaviors are affected by the surface adsorption of probe particles, actin liquid crystalline (LC) phase transition, and counterion mediated interaction between filaments. An introduction to actin, F-actin phase transition, counterion mediated attraction between like-charged filaments, and microrheology of isotropic actin solutions is provided in Chapter 1. Chapter 2 describes the ex- perimental techniques used in the study including various microscopic methods and single particle tracking microrheology. In Chapter 3, we find the measured microrhe- ological properties of actin solution dependent on the probe size and probe surface coating, and we propose that hopping and surface adsorption account for two op- posite trends of probe size dependence of moduli measured by sticky and slippery particles, respectively [1]. In Chapter 4, we report anisotropic viscoelastic properties vii of nematic F-actin solutions by extending the particle tracking microrheology tech- nique to anisotropic materials, and the effect of divalent counterion on the mechanical properties has also been compared for isotropic and nematic solutions. These results were published in Ref. [2]. In Chapter 5, we report an abnormal slowdown of F-actin diffusion across the isotropic to nematic phase transition reThis thesis provides a mi- croscopic investigation of mechanical properties and dynamics of filamentous actin (F-actin) networks via micron-sized particle tracking microrheology and single fila- ment tracking technique, respectively.gion, and the abnormal slowdown is proposed to be caused by the weak electrostatic attraction between actin filaments mediated by the divalent counterion in the nematic phase, with the results published in Ref. [3]. Finally, a summary of this thesis is given in Chapter 6. Jun He viii Acknowledgments I would like to express special and tremendous thanks to my thesis advisor, Prof. Jay X. Tang, for his invaluable guidance, inspiration, encouragement throughout my thesis work, as well as his sustained efforts to train me to be a scientist. I will always cherish the experience working with him. I would like to acknowledge all the current and former lab members, including: Dr. Guanglai Li, Dr. Karim Addas, Dr. Jorge Viamontes, Dr. Qi Wen, Dr. Yongxing Guo, Dr. Yifeng Liu, Hyeran Kang, Patrick Oakes, Michael Mak and Jingjing Wang. Among them, I especially want to thank Dr. Karim Addas for helping with the optical tweezers setup; Dr. Yongxing Guo for insightful discussion of my first paper, and helping with lyx, LaTex and MatLab; Hyeran Kang and Jingjing Wang for being my last year office-mates with pleasure. Special appreciation also goes to Prof. Alex Levine, Prof. Christoph F. Schmidt, Prof. Margaret Gardel and Prof. Megan Valentine, for their valuable suggestions to our work through discussions or email communications. I would also thank the thesis committee members, Prof. James Valles and Prof. Thomas Powers, for their great help and precious advice towards the thesis. ix I want to thank my parents, for their unconditional support and endless encour- agement throughout my growth, without which I would not have come this far. I would also thank my friends at Brown, including Hua Li, Feifei Li, Chenjie Wang, Wenzhe Zhang, Chao Gong, Kongbin Kang, for the good time we had together. Spe- cial thanks go to Feifei Li, my classmate for 12 years, for his help and advice in many aspects. Last but not least, I would like to thank my girlfriend Xian Luo for her companionship, love, and support throughout my Ph.D endeavor. x Contents Vita iv Preface vii Acknowledgments ix List of Figures xvii 1 Introduction 1 1.1 Actin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Molecular structure . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Monomeric actin polymerization to filamentous actin . . . . . 4 1.1.3 Structural and mechanical properties of actin filaments . . . . 8 1.2 Isotropic to Nematic Liquid Crystalline Phase Transition of F-actin Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Isotropic to nematic liquid crystalline phase transition of rod- like systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 xi 1.2.2 Isotropic to nematic liquid crystalline phase transition of F- actin network . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Counterion Mediated Electrostatic Interaction and Actin Bundle For- mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.1 Counterion condensation . . . . . . . . . . . . . . . . . . . . . 14 1.3.2 Counterion mediated like-charge attraction . . . . . . . . . . . 16 1.3.3 Actin bundle formation by like-charge attraction . . . . . . . . 17 1.4 Physical Effects on Microrheology of F-actin Network . . . . . . . . . 20 1.4.1 Microrheology of isotropic F-actin networks . . . . . . . . . . 20 1.4.2 Physical effects on microrheology of F-actin networks . . . . . 21 1.4.2.1 Mesh size and probe size . . . . . . . . . . . . . . . . 21 1.4.2.2 Surface adsorption . . . . . . . . . . . . . . . . . . . 22 1.4.2.3 Depletion effect . . . . . . . . . . . . . . . . . . . . . 24 1.4.2.4 Probe size effect . . . . . . . . . . . . . . . . . . . . 24 1.4.2.5 Isotropic to nematic phase transition . . . . . . . . . 25 2 Experimental Methods 26 2.1 Sample preparation of F-actin solution . . . . . . . . . . . . . . . . . 26 2.2 Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.1 Fluorescence microscopy . . . . . . . . . . . . . . . . . . . . . 27 2.2.2 Phase contrast microscopy . . . . . . . . . . . . . . . . . . . . 28 2.2.3 LC-Polscope system . . . . . . . . . . . . . . . . . . . . . . . 33 xii 2.3 Microrheology Techniques . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.1 Rheology and microrheology . . . . . . . . . . . . . . . . . . . 35 2.3.2 Active microrheology . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.3 Passive microrheology . . . . . . . . . . . . . . . . . . . . . . 37 2.3.3.1 Video particle tracking (VPT) microrheology . . . . 37 2.3.3.2 Laser deflection particle tracking (LDPT) microrheol- ogy . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.3.3 VPT vs. LDPT methods . . . . . . . . . . . . . . . . 43 2.3.4 Two particle Microrheology . . . . . . . . . . . . . . . . . . . 44 3 Surface Adsorption Causes an Opposite Probe Size Dependence to that of Hopping in Microrheology of Actin Network 47 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.1 Preparation of BSA coated PS beads . . . . . . . . . . . . . . 51 3.2.2 Preparation of PEG grafted PS beads . . . . . . . . . . . . . . 51 3.2.3 Sample preparation for microrheology . . . . . . . . . . . . . 53 3.2.4 Preparation of Alexa-488 labeled actin for confocal imaging . . 53 3.2.5 Confocal imaging of beads and actin network . . . . . . . . . 54 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3.1 Direct comparison of protein adsorption by confocal images . 56 3.3.2 Physisorption leads to size dependent viscoelasticity measurement 57 xiii 3.3.3 Cage hopping detected using adsorption resistant beads smaller than the network mesh size . . . . . . . . . . . . . . . . . . . 60 3.3.4 Opposite trends of scaled MSD vs. probe size for adsorption and non-adsorption surface chemistries . . . . . . . . . . . . . 64 3.3.5 Opposite trends of moduli vs. probe size for adsorption and non-adsorption probes . . . . . . . . . . . . . . . . . . . . . . 65 3.3.6 Depletion effect detected from BSA coated beads . . . . . . . 67 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4 Counterion Dependent Microrheological Properties of F-actin Solu- tion across Isotropic-Nematic Phase Transition 78 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.2 VPT and LDPT microrheology . . . . . . . . . . . . . . . . . 81 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3.1 Effects of probe beads on nematic ordering . . . . . . . . . . . 82 4.3.2 Anisotropic bead diffusion in the nematic phase . . . . . . . . 84 4.3.3 Mean square displacement and the bead size effect . . . . . . . 87 4.3.4 Frequency spectrum of shear moduli of nematic F-actin solutions 89 xiv 4.3.5 Shear modulus versus actin concentration across the isotropic- nematic phase transition . . . . . . . . . . . . . . . . . . . . . 92 4.3.6 Dependence of shear moduli on the Mg2+ concentration in isotropic and nematic phases . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.4.1 Anisotropy of viscoelastic properties of nematic F-actin networks 96 4.4.2 Dependence of shear moduli on actin concentration for nematic F-actin solutions . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.4.3 Viscoelasticity of F-actin solutions altered by Mg2+ concentration101 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5 Counterion Induced Abnormal Slowdown of F-actin Diffusion across Isotropic to Nematic Phase Transition 105 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.2.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . 108 5.2.2 Single filament tracking method . . . . . . . . . . . . . . . . 109 5.2.3 Birefringence measurement using Polscope . . . . . . . . . . . 111 5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.3.1 Single filament reptation in a virtual tube . . . . . . . . . . . 111 5.3.2 Abnormal slowdown of filament diffusion across the isotropic to nematic phase transition . . . . . . . . . . . . . . . . . . . . . 114 xv 5.3.3 Counterion mediated temporary association accounts for the slowdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.3.4 A simple model . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6 Summary of Thesis 124 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 xvi List of Figures 1.1 Ribbon representation of the structure of uncomplexed actin in the ADP state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Actin polymerization and ATP hydrolysis . . . . . . . . . . . . . . . . 5 1.3 Structure of actin filament . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Isotropic to nematic phase transition of F-actin solution . . . . . . . . 11 1.5 Actin bundle formation and like-charge attraction . . . . . . . . . . . 18 2.1 Principle of excitation and emission for use in fluorescence microscopy 29 2.2 Reflected light fluorescence microscopy . . . . . . . . . . . . . . . . . 30 2.3 Phase contrast microscopy . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4 LC-Polscope configuration for an upright microscope . . . . . . . . . 34 2.5 Experimental setup for Laser deflection particle tracking (LDPT) method 41 3.1 Confocal images of Alexa-488 labeled F-actin solutions embedded with different beads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 xvii 3.2 MSDs and scaled MSDs of BSA coated PS beads in a 0.34 mg/mL F-actin solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 The frequency spectra of storage and loss moduli of a 0.34 mg/ml F-actin solution measured by different sized BSA coated PS beads . . 59 3.4 MSDs and scaled MSDs of PEG coated PS beads in a 0.34 mg/mL F-actin solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.5 The frequency spectra of storage and loss moduli of 0.34 mg/ml F-actin solution measured by different sized PEG coated PS beads . . . . . . 62 3.6 The scaled MSDs as functions of bead diameter for four types of beads 63 3.7 The storage and loss moduli as functions of bead diameter for four types of beads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.8 Radially averaged intensity profile from the center of the bead for BSA coated beads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.9 A schematic illustration of hopping and surface adsorption . . . . . . 70 4.1 Birefringence images of nematic actin solution with beads embedded . 83 4.2 Trajectories and distributions of bead positions in parallel and perpen- dicular directions for isotropic actin solutions . . . . . . . . . . . . . . 85 4.3 Trajectories and distributions of bead positions in parallel and perpen- dicular directions for nematic actin solutions . . . . . . . . . . . . . . 86 4.4 MSD of 1 µm beads as a function of time interval in isotropic and nematic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 xviii 4.5 Frequency dependence of G0 and G00 measured by the VPT method for isotropic F-actin solutions . . . . . . . . . . . . . . . . . . . . . . . . 89 4.6 Frequency dependence of G0 and G00 measured by the VPT and LDPT methods for nematic F-actin solutions . . . . . . . . . . . . . . . . . . 91 4.7 Dependence of G0 and G00 on actin concentration measured by the VPT method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.8 G0 and G00 in the parallel and perpendicular directions as a function of [M g 2+ ] measured by the VPT method . . . . . . . . . . . . . . . . . 95 4.9 Illustrations of beads embedded in the isotropic and nematic actin networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.1 Illustration of the reptation tube model for F-actin . . . . . . . . . . 107 5.2 Procedure of single filament tracking . . . . . . . . . . . . . . . . . . 110 5.3 MSD and diffusion coefficient of different filament lengths . . . . . . . 112 5.4 Microscopic viscosity and specific retardance as a function of actin concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5 Microscopic viscosity as a function of normalized concentration for dif- ferent macromolecules . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.6 Microscopic viscosity vs. Magnesium concentration . . . . . . . . . . 120 xix Chapter 1 Introduction 1.1 Actin Actin is one of the most abundant and highly conserved eukaryotic proteins, with a globular form and a molecular weight of 42 kDa. Actin is one of the three major cytoskeletal components found in almost all eukaryotic cells [4]. Complemented by other associated proteins, a dynamic actin network serves a variety of cell functions, such as motility, division, cell shape change and mechanoprotection [5]. Actin is highly conserved throughout evolution, with 80.2% sequence conservation at the gene level between human and yeast, and 95% conservation of the primary structure of the protein product [6]. Most yeasts have only a single actin gene; higher eukaryotes, however, express several isoforms of actin encoded by a family of related genes. For instance, mammals possess at least six actin isoforms encoded by separate 1 2 genes [7], which are divided into three classes according to their isoelectric point, i.e. alpha, beta and gamma. Generally, alpha actins are present in muscle, whereas beta and gamma isoforms are mostly found in non-muscle cells. Different isoforms of actin have highly similar amino acid sequences and in vitro properties, yet, they cannot completely replace each other for in vivo functions [8]. In this section, I will give an introduction of background knowledge about actin, including a brief history, protein structure, actin polymerization and hydrolysis, struc- tural and mechanical properties of filamentous actin. 1.1.1 Molecular structure The discovery of actin was credited to Brúnó F. Straub, a biochemist working in the Institute of Medical Chemistry at the University of Szeged, Hungary. In 1942, Straub developed a novel technique for extracting muscle protein that allowed him to isolate substantial amounts of relatively-pure actin. He extracted actin with water from the acetone-treated and dried residue of muscle tissue after myosin had been separated [10], or from actomyosin similarly treated with acetone and dried [11]. Straub’s method is essentially the same as that used in laboratories today. Since Straub’s protein was initially reported to cause myosin to appear as an “activated” viscous form, it was named actin. The complete amino acid sequence of rabbit muscle actin was resolved by Elzinga et al. [12]. Later, the three-dimensional atomic structure of actin complexed with 3 Figure 1.1: Ribbon representation of the structure of uncomplexed actin in the ADP state adopted from Ref. [9]. The four subdomains of actin are represented in different colors: subdomains 1 (purple), 2 (green), 3 (yellow), and 4 (red). ADP is bound at the center of the molecule where the four subdomains meet. Four Ca2+ ions bound to the actin monomer in the crystals are represented as gold-colored spheres. Polymerization was blocked by covalent binding of the fluorescent probe tetramethylrhodamine-5-maleimide (TMR) to Cys374. 4 deoxyribonuclease I, gelsolin segment, or profilin in the ATP or ADP state was de- termined by X-Ray analysis [13, 14, 15]. The actin molecule consists of two domains which each can be further divided into two subdomains. ADP or ATP is located in the cleft between the domains with a calcium ion bound to the β−, or β− and γ− phosphates, respectively. More recently, Ludovic et al. resolved the structure of uncomplexed actin in the ADP state using X-ray crystallography [9]. Compared with previous ATP-actin structures in the complexed form, monomeric ADP-actin is characterized by a marked conformational change in subdomain 2. This conforma- tional difference suggests that nucleotide-dependent differences in this location may provide a mechanism to change the orientations of the actin subdomains relative to each other, and hence affect the dynamics of actin polymerization. Figure 1.1 shows a ribbon representation of the structure of uncomplexed actin in the ADP state adopted from Ref. [9]. 1.1.2 Monomeric actin polymerization to filamentous actin Actin polymerization and depolymerization are necessary in cell migration and cy- tokinesis. Monomeric actin (G-actin) polymerizes to form filamentous actin (F-actin) upon addition of salts at physiological concentrations (for instance, 50 mM KCl and 2 mM MgCl2 ). Polymeric F-actin is formed by reversible noncovalent self-association of monomeric G-actin. Straub was the first to study actin polymerization, and in 1950 he found that the polymerization from G-actin to F-actin (G-F transformation) 5 Figure 1.2: Illustrations to show the dynamic process of actin polymerization and accompanied ATP hydrolysis. (a) An actin monomer has two domains hinged around a ATP binding site. The bound ATP hydrolyzed immediately after the monomer is incorporated to the filament. The resulting ADP is trapped in the monomer and can be exchanged with ATP only after depolymerization, since the two domains are hold together by the interatction with neighboring actin subunit. (The drawing is adopted from [4]) (b) Dynamic process of actin polymerization. Both ends can have actin monomer association and dissociation. In the steady state, actin filament assembles at the plus end and disassembles at the minus end simultaneously. ATP hydrolysis happens right after the actin monomer is attached to the filament, transforming actin from ATP-bound state to ADP-bound state. The Pi is still assicated with actin for certain time, and ADP.Pi state stabilize the filament. Pi release destabilizes the filament, promoting depolymerization. 6 involves ATP hydrolysis [16]. G-actin polymerizes to form F-actin above a critical concentration (Cc ) of G-actin with presence of K+ , Mg2+ ions and ATP. Polymerized F-actin coexist with active G-actin, the concentration of which is independent of the F-actin concentration and equal to the G-actin concentration at the critical point [17]. Self-polymerization of actin is a process having a lag phase followed by a pseudo-first-order decay process, suggesting that actin polymerization consists of distinct nucleation and elongation phases [18]. The lag in the polymerization is due to the kinetic barrier, and the nucleation rate is proportional to the cubic power of actin concentration, indicating a nucleating structure of trimers; in contrast, the elongation rate is proportional to free actin subunit concentration, suggesting that filaments elongate by addition of one actin at a time [4]. ATP hydrolysis accompanies actin polymerization. ATP bound in G-actin is hy- drolyzed immediately after the G-actin is incorporated into an actin filament. ATP hydrolysis during polymerization transforms actin from the ATP-bound state to the ADP-bound state, releasing an inorganic phosphate to solution. The ADP is trapped within the actin monomer incorporated in the filament until the monomer is depoly- merized, and then exchanges its ADP with the ATP in solution, returning to the ATP-bound state to undergo polymerization again [4]. Detailed experimental and modeling studies [19, 20, 21] have revealed that ATP hydrolysis is not absolutely nec- essary for polymerization. Also, the hydrolysis reaction during actin polymerization 7 occurs in two steps: cleavage of ATP followed by the slower release of the inorganic phosphate (Pi). Consequently, a cap of ADP · Pi-actin subunits (in the steady state) or a transient cap of ATP-actin (in the high growing state) exists at the barbed end of a filament, stabilizing the filament. The release of Pi destabilizes the filament. The role of ATP hydrolysis is illustrated in Fig. 1.2. In sum, ATP hydrolysis is not re- quired for actin polymerization, but is necessary to promote F-actin depolymerization (after Pi is released). Actin polymerization is polarized [4]. In the growing phase, the two ends of actin filaments polymerize at different rates: the fast growing end is called the plus end (barbed end); the slow growing end is called the minus end (pointed). Each end has its own association rate, kon , and dissociation rate, kof f . Even though the rate constant kon and kof f are up to 10 times larger at the plus end than the minus end, the ratio kof f /kon (or Cc ) is the same at both ends if ADP is the only nucleotide ADP AT P available. In the presence of ATP, however, its hydrolysis makes Cc = kof f /kon at the two ends different, with Cc (minus) > Cc (plus). Net polymerization proceeds until the free monomer concentration falls between the Cc values of the plus and minus ends, respectively. In this equilibrium phase, the filament assembles at the plus end and disassembles at the minus end simultaneously, while the filament main- tains its constant length, known as treadmilling [22]. The dynamic process of actin polymerization is shown in Fig. 1.2b. 8 Figure 1.3: (a) Electron micrograph of negatively stained actin filaments with a scale bar of 50 nm. (b) Schematic of the arrangement of monomers in a filament. (a) and (b) are adopted from Ref. [4]. (c) Atomic model structure of actin filament, adopted from Ref. [23]. 1.1.3 Structural and mechanical properties of actin filaments F-actin is composed of two protofilaments twisted around one another to form a right handed double helix, with 37 nm translation per 1/2 turn (Fig. 1.3b). The diameter of F-actin is measured to be 8 nm [24, 23]. An atomic model of the actin filament has been constructed by Holmes et al. [23] based on the atomic structure of actin monomer by fitting X-ray fiber diffraction data from oriented F-actin gels (Fig. 1.3c). Due to the secondary bonding between the actin monomers and the topology of the filament, F-actin is a flexible polymer. Flexibility can be quantified by the persis- tence length, which is defined as the length over which correlations in the tangential directions are lost. The persistence length is related to the elastic property of the 9 filament [25] as Lp = E/kB T , where E is the bending modulus of the filament in the unit of Energy×Length, kB is the Boltzmann constant, and T is the absolute tem- perature. An intuitive understanding of persistence length is the distance traveled on a filament bent due to thermal fluctuations such that the tangent angle changes by 1 radian. For extremely stiff biopolymers such as microtubules, the persistence length is on the order of mm, whereas for DNA, which is quite a flexible polymer, the persistence length is approximately 100 nm. The persistence length of F-actin can be obtained by studying the bending dynamics under thermal fluctuations, first observed with dark field microscopy by Nagashima in 1980, who decorated the F-actin with myosin subfragments to increase the filament width into the visible light microscope regime of 200 nm [26], and later revolutionized by introduction of fluorescence mi- croscopy. Many have since measured the persistence length of F-actin to be 15-18 µm [27, 28, 29]. 1.2 Isotropic to Nematic Liquid Crystalline Phase Transition of F-actin Solution As a rod-like system, F-actin solution undergoes an isotropic to nematic liquid crys- talline phase transition when the protein concentration increases above certain thresh- old value. This phase transition has been extensively studied [30, 31, 32, 33, 34, 35]. In Chapter 4 and Chapter 5, I will discuss our work on the counterion dependent 10 microrheological properties and F-actin diffusion dynamics across the isotropic to nematic phase transition, respectively. In this section, I will first provide the necessary background knowledge about the isotopic to nematic liquid crystalline phase transition of rod-like polymer systems in general, including basic concepts and models. Then, I will present a survey of literature for the isotopic to nematic phase transition of F-actin solutions on nematic order parameter, phase diagram, and order of F-actin phase transition. 1.2.1 Isotropic to nematic liquid crystalline phase transition of rod-like systems For systems composed of rod-like molecules, nematic phase is one of the most common liquid crystalline (LC) phases, which also include cholesteric, smectic, and columnar phases [36]. The isotropic (I) phase possesses neither positional nor orientational order of the molecules. The nematic (N) phase is the least ordered liquid crystal phase, with only long range orientational order but no positional order of the molecules. The molecules flow in a liquid manner with their center of mass positions randomly distributed, while they point in the same direction. The preferential orientation is called the nematic director. The order parameter of the nematic phase is defined as R S = f (θ)P2 (θ)dΩ, where f (θ) is the orientational distribution function and P (θ) is the second Legendre polynomial. Liquid crystals can be divided into two types: thermotropic and lyotropic. Thermotropic LCs undergo a phase transition mainly 11 Figure 1.4: Isotropic (I) to nematic (N) phase transition of F-actin solution. (a) F- actin solution in the isotropic phase. Fluorescently labeled actin filaments embedded in unlabeled backgroud solution with a low concentration of 0.2 mg/ml show neither alignment nor positional order. (b) F-actin solution in the nematic phase. Fluores- cently labeled actin filaments embedded in an 8 mg/mL unlabeled backgroud solution show preferential alignment without positional order. (a, b adopted from Ref. [34]) (c) Birefringent retardance of actin solution as a function of actin concentration and the onset concentration is higher for shorter average filament lengths. Specific re- tardance is proportional to order parameter. (d) Birefringence measurements of an F-actin solution with an average filament length of 1 µm in the I-N transition region. F-actin phase separates into nematic tactoidal droplets in an isotropic background (c, d adopted from Ref. [35]) (e) A phase diagram of F-actin I-N phase transition com- pared with Flory’s lattice model (solid line) obtained by Suzuki et al. [30]. Isotropic solutions are in open symbols; nematic solutions are closed symbols. 12 depending on temperature, whereas lyotropic LCs exhibit phase transitions primarily as a function of concentration [36]. Lars Onsager in 1949 proposed a statistical model of hard-rods with steric inter- actions, which predicts lyotropic LC phase transitions [37]. This theory considers the volume excluded from the center-of-mass of one idealized cylinder as it approaches another. Based on this model, Onsager predicted that the volume fraction of rods to total volume just at the onset of I-N coexistence region is φI = 3.34 (D/L), where D and L are the diameter and length of the rod, and the volume fraction of the nematic phase at the transition point is φN = 4.49 (D/L) [37]. P. J. Flory in 1956 developed a lattice model which is also a steric theory [38]. The volume of material (rods and solvent) is divided into cubic lattice cells. If the cell holds part of a rod, it is occupied. Otherwise, the cell is empty. All rods are initially aligned with the nematic director. When the rod tilts an angle, it is represented with segments parallel to the director. Flory’s lattice model shows that φI = 8 (D/L) and φN = 12.5 (D/L). Flory’s lattice model gives an exact solution when all the rods are parallel, while Onsager’s hard-rod model describes the isotropic phase better. In this sense, Flory’s and Onsager’s models complement each other. These model results imply that the phase transition of lyotropic LC materials depends on the concentration and aspect ratio of the molecules. 13 1.2.2 Isotropic to nematic liquid crystalline phase transition of F-actin network F-actin undergoes an isotropic (I) to nematic (N) liquid crystalline phase transition as a function of filament concentration and average filament length. Figure 1.4a shows fluorescently labeled actin filaments randomly oriented in an unlabeled isotropic background; Figure 1.4b shows labeled actin filaments aligned approximately in the same direction in an unlabeled nematic solution. The onset concentration of the I- N phase transition of F-actin is inversely proportional to the average filament length [30, 31, 32, 33, 34, 35] (Figure 1.4c), consistent with statistical models discussed in the previous subsection [37, 38]. The order parameter of nematic actin solution increases with actin concentration, and saturates at about 0.75 [34], indicating that high extent of entanglement of actin filaments prevents them from perfect alignment. A phase diagram of F-actin I-N phase transition compared with Flory’s lattice model is shown in Figure 1.4e [30]. The F-actin I-N phase transition can be a continuous or first order phase transition depending on the average filament length. For average filament lengths longer than 2 µm, the F-actin I-N transition appears to be continuous in both filament alignment and concentration [31, 33]. Long filament length, polydispersity, and semiflexibility of the filaments may contribute to entangling the network, ultimately resulting in a continuous transition [33]. For average filament lengths shorter than 2 µm, F-actin 14 I-N transition has been demonstrated to be a first order transition with a clear discon- tinuity in both alignment and concentration. Both nucleation growth and spinodal decomposition were observed (Fig. 1.4d) [35]. 1.3 Counterion Mediated Electrostatic Interaction and Actin Bundle Formation In this section I will review the basic concepts of counterion condensation, two the- oretical models for counterion mediated like charge interaction, and finally, a survey of literature on multi-valent counterion induced actin bundle formation. Concepts of counterion condensation and like-charge attraction provide mechanisms by which to explain the effect of divalent counterions on the microrheological properties of F- actin solution (Chapter 4), and the abnormal slowdown of F-actin diffusion across the isotropic to nematic phase transition (Chapter 5). 1.3.1 Counterion condensation Polymers with repeating units bearing an electrolyte group are called polyelectrolytes. Polyelectrolytes become charged when these groups dissociate in aqueous solutions. Most biopolymers, such as DNA and F-actin, carry certain amounts of structural charge due to the ionizable residues in their monomeric subunits. DNA strands and ◦ F-actin carry negative charges with linear charge densities of -e/1.7 A [39] and -e/2.1 15 ◦ A [40], respectively. Counterions condense into a thin layer close to the charged polyelectrolyte surface. Within this condensation layer, the counterion concentration is much higher than that in the bulk solution. This concept of counterion condensation was developed inde- pendently by G. Manning [41] and F. Oosawa [42] in the 1960’s, and is also referred to as the Manning-Oosawa condensation theory. Manning demonstrated that a highly charged cylinder-like polymer exerts a strong attraction on the counterions causing a certain fraction of them to condense onto the polymer in the limit of infinitely thin polymers and infinitely low salt concentration. Oosawa applied a two-state model which predicted a condensed and gas-like counterion phase near the polyelectrolyte surface. They both predicted counterion condensation, when the manning parame- ter ξ = lB /l is greater than 1, where e/l is the backbone linear charge density and lB = e2 /4πεε0 kB T is the Bjerrum length. The counterions are condensed around the polyelectrolyte to reduce the effective linear charge density until ξ = 1 [43, 44]. The ◦ Bjerrum length of water is lB = 7.13 A at 25 ◦ C, whereas for DNA and F-actin l =1.7 ◦ and 2.1 A, respectively. Thus, counterion condensation is expected to occur to both macromolecules. 16 1.3.2 Counterion mediated like-charge attraction Like-charged polyelectrolytes attract each other with presence of counterions under certain conditions, which is referred to as like-charge attraction. Like-charge attrac- tion is observed for a variety of biopolymer systems. DNA strands attract each other to form aggregates due to like-charge attraction induced by multivalent counterions, which is often referred to as DNA condensation [45, 46, 47]. DNA condensation is crucial for packing high volume of genetic information in the cell nucleus. F-actin also forms finite sized bundles in the presence of multi-valent counterions [48, 46, 40]. Actin bundles are responsible for various cellular functions like motility, cytokinesis and shape change [5]. Two models based on different types of counterion correlations describe the physi- cal mechanism which causes an attractive interaction between like-charged polymers, namely, the Oosawa model [42] and the Wigner crystal model [49, 50]. The Oosawa model states that the counterions condensed onto the charged polyelectrolyte are free to move within the condensation layer and exchange with the bulk solution. Ther- mal fluctuations induce the counterion density fluctuations along a charged polymer, resulting in transient alternating regions of high and low counterion densities. When two parallel polyelectrolytes come close, they form transiently complementary coun- terion density profiles: the low density regions of one polyelectrolyte correspond to high density regions of the other, creating an attractive interaction. This mechanism is analogous to van der Waals interactions between atoms and molecules [42, 51]. 17 The Wigner crystal model predicts a like-charge attractive interaction based on positional correlations of counterions [49, 50]. The model treats the condensed coun- terions as a one component plasma, or a strongly correlated liquid, since the Coulomb interaction between individual counterions is much higher than the thermal energy kB T due to the short distance between condensed counterions. At a low temperature, this strong electrostatic correlation causes counterions to form a Wigner lattice on the polyelectrolyte surface. The cohesive energy of the Wigner crystal creates an attractive interaction when the distance between polyelectrolytes is comparable to the lattice constant of the Wigner crystal. Counterions on the two polyelectrolytes correlate with each other to form a new Wigner crystal with a higher density of ions and lower free energy. 1.3.3 Actin bundle formation by like-charge attraction F-actin is observed to form bundles induced by a number of polycations including multi-valent metal ions, Co(NH3 ), and basic polypeptides [48, 46]. Fig. 1.5a shows a fluorescence image of actin bundles formed by adding Mg2+ ions. The general features of bundle formation are largely independent of the specific structure of the bundling agent used. Actin bundle formation requires a threshold polycation concentration, which varies strongly with the valence of the cation and increases with the ionic strength of the solution (See Fig. 1.5b). Most features of actin bundle formation by polycation can be explained by the counterion condensation theories [42, 49, 51]. 18 Figure 1.5: Actin bundle formation and like-charge attraction. (a) Fluorescence image of actin bundles formed from a 0.1 mg/ml actin solution induced by 50 mM M gCl2 . (b) Light scattering signal of F-actin as a function of concentration of various cations (adopted from Ref. [48]). Each sample contained initially 0.5 mg/ml F- actin at pH 7.2, followed by sequential additions of concentrated cations. Different cations induce bundle formation at different critical concentrations. (c) Schematic representations of uncondensed and condensed F-actin (adopted from Ref. [40]). Left: At low multivalent ion concentrations, two F-actin filaments maintain their native symmetry and are unbound. Right: At high ion concentrations, the ions collectively form a charge density wave, which bundles the F-actin filaments, and moreover forms a coupled mode with torsional distortions of the F-actin and has overtwisted it by 3.8◦ per monomer. 19 It has recently been observed by X-ray diffraction that counterions organize into ‘‘frozen’’ ripples parallel to the actin filaments and form 1D charge density waves, which attract the filaments to form bundles [40] (See Fig. 1.5c). A generic feature of actin bundle formation by polycations is that the bundles observed in experiments are always of finite size [52]. This contradicts theoretical descriptions of polyelectrolyte condensation, which predict infinitely large aggregates in equilibrium [49, 53]. To resolve this puzzle, it has been suggested that the growth kinetics of actin bundles play a role [53], though later experiments demonstrated that kinetics play a role in bundle growth but not in the lateral size of bundles [52]. Alternatively, it has been proposed that steric and short-range electrostatic interactions due to the finite size of counterions prevent charge neutralization of the bundle, thus forcing the equilibrium bundle size to be finite [54]. More recent X-ray diffraction experiments reported that size control relies on a mismatch between the helical structure of individual actin filaments and the geometric packing constraints within bundles [55]. 20 1.4 Physical Effects on Microrheology of F-actin Net- work Microrheology is a powerful tool to probe the microscopic mechanical properties of polymeric materials, biomaterials and cells. Microrheology methods extract viscoelas- tic properties of material from the response of embedded micron-sized probes under the influence of external force (active methods) or thermal excitations (passive meth- ods). In this section, I will give a brief review of the literature of actin microrheology, and the physical effects that may affect microrheology results. Details about mi- crorheology techniques are deferred to Chapter 2. 1.4.1 Microrheology of isotropic F-actin networks Viscoelastic properties of actin network have been studied extensively over the last decade or two using different single particle microrheology techniques (passive and active), such as magnetic tweezers [56, 57], optical tweezers [58], diffusing wave spec- troscopy (DWS) [59, 60, 61, 62], laser deflection particle tracking (LDPT) [63, 64] and video particle tracking (VPT) [65, 66, 67, 68] methods, as well as two particle techniques [69, 66, 68]. The storage modulus of a pure F-actin solution is reported to weakly depend on concentration as G0 (c) ∼ c1.2 [60] or G0 (c) ∼ c1.8 [66], and both the storage G0 (ω) and loss moduli G00 (ω) scale with frequency with an exponent of 0.75 at high frequencies, i.e. ∼ ω 3/4 [63, 60, 61]. This scaling behavior of the moduli is 21 also predicted by theoretical modeling, considering the dominance of single-filament dynamics at frequencies ω>100 radian/second [70]. A two particle cross-correlation method has been compared with single particle methods to be less sensitive to the inhomogeneity of actin network and reflect the true bulk behavior better [69, 66]. Almost all the previous studies on actin microrheology were focused on the isotropic F-actin networks, and few addressed actin solutions in the nematic phase. In Chap- ter 4, we extend single particle microrheology methods to nematic actin networks, characterizing the viscoelastic properties parallel and perpendicular to the nematic director. We also study how the mechanical properties of actin network are affected by the divalent counterion concentration differently in the isotropic phase and nematic phase. 1.4.2 Physical effects on microrheology of F-actin networks 1.4.2.1 Mesh size and probe size If chemical interactions between probe particles and networks are negligible, the mo- tion of probe particle can be characterized based only on the probe size relative to the mesh size of the network. When thermally driven embedded probe particles are large compared to the network mesh size, the ensemble-averaged MSD is directly related to the frequency-dependent storage and shear moduli using a generalized Stokes-Einstein relation [71, 64, 72]. On the contrary, when the probe particles are approximately equal to or smaller than the network mesh size, the particles make infrequent jumps 22 between distinct pores or percolate through the network, and hence their dynamics are no longer dictated by the bulk viscoelastic response [73]. Rather, their thermal motions are sensitive to the viscosity of the solvent, the effects of macromolecular crowding, and steric and hydrodynamic interactions with the network [74]. Since many biopolymer networks are structured on length scales of micron, even small changes in the probe size or the network mesh size may cause big differences in the microrheology measurements. 1.4.2.2 Surface adsorption Chemical interactions between probe particles and networks may introduce large am- biguities in the interpretation of network responses. For microrheology measurements, the particles must be sufficiently resistant to protein adsorption in order to prevent the local modification of network architecture and introduction of small heterogeneities. For instance, when the probe particles are smaller than the network mesh size while probing microenvironments of a heterogeneous material, even a small amount of pro- tein adsorption can cause particles to adhere to cavity walls, preventing them from fully exploring small pores and possibly even inducing local changes in network struc- ture [67]. McGrath et al. [65] demonstrated that for both crosslinked and un-crosslinked actin networks, particles that bind F-actin report viscoelastic moduli comparable to those determined by macroscopic rheology experiments, yet, particles modified to 23 prevent actin binding detect weaker microenvironments. They tested bovine serum albumin (BSA) coated polystyrene (PS) beads, carboxylate (COO− ) PS beads, bare PS beads, silica beads, NH3 −PS beads, and polylysine coated PS beads with progres- sively stronger protein adsorption, and found a positive correlation between surface stickiness and measured moduli. Even when adjacent in the same cross-linked gel, actin-binding particles report viscoelastic moduli two orders of magnitude higher than nonbinding particles at low frequencies (0.5∼1.5 rad/sec) but the difference converges at high frequencies (104 rad/sec or above). Valentine et al. [67] developed a simple protocol to prepare polyethylene glycol (PEG) grafted probe particles which adsorb significantly less protein than particles coated with BSA or unmodified probe particles. They found that varying particle surface chemistry selectively tunes the sensitivity of the particles to different physical properties of their microenvironments, and specifically, particles that weakly bind to a heterogeneous network are sensitive to changes in network stiffness, whereas protein- resistant tracers detect changes in the viscosity of the fluid and in the network mesh size. Two particle microrheology measurements eliminate the differences arising from surface chemistry. One consensus from the previous studies is that slight protein ad- sorption of probe surface is required for microrheology to best reflect bulk rheological properties [65, 67]. 24 1.4.2.3 Depletion effect Recently, Fisher et al. reported a depletion of actin filaments near a BSA coated flat glass surface using confocal microscopy and fluorescence intensity analysis [75]. This depletion effect can be visualized as the rigidity of F-actin hindering a filament’s approach to a solid surface. Similar steric interactions between a probe particle and its surrounding network are also expected to cause such an effect, creating a depletion layer around a probe particle with lower filament density than in the bulk, which may affect microrheology measurements. In Chapter 3, we confirm the existence of a depletion layer around BSA coated PS bead using a technique similar to that recently used by Fisher et al. [75], and qualitatively assess its effect on microrheology. 1.4.2.4 Probe size effect Ideally, the shear moduli measured by microrheology is independent of the probe size. In experiments, the measured moduli may depend on probe size due to the ratio of probe size to mesh size, the surface adsorption effect and the depletion effect. To our knowledge, the probe size dependency of microrheology has not been studied systematically. In Chapter 3, we study how these three factors affect the dependence of the shear moduli measured by microrheology. 25 1.4.2.5 Isotropic to nematic phase transition Many groups have studied viscoelastic properties of conventional nematic polymer liquid crystals using different rheology techniques [76, 77, 78, 79]. However, among them, few covered the anisotropic behavior of shear modulus 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 temperature of elastomers and poly- mers [80, 81, 82]. Martinoty et al. [82] showed the anisotropic viscoelasticity of nematic elastomers, with G0⊥ larger than G0k . As for the microrheology studies of biopolymer network, almost all studies have focused on the isotropic networks. In Chapter 4, we extend single particle microrheology methods to the nematic actin network, characterizing the viscoelastic properties parallel and perpendicular to the nematic director. We also study how the mechanical properties of actin networks are affected by the divalent counterion concentration differently in the isotropic phase and nematic phases. Chapter 2 Experimental Methods 2.1 Sample preparation of F-actin solution Actin was extracted from rabbit skeletal muscle following the technique of Pardee and Spudich [83]. The extracted actin, kept in G-buffer (2 mM Tris-HCl, pH 8.0, 0.5 mM ATP, 0.2 mM CaCl2 , 0.5 mM DTT, and 0.005% NaN3 ), was frozen with liquid nitrogen and stored at −80 ◦ C. For experiments, an aliquot of actin was thawed rapidly on a heatblock to 25 ◦ C and centrifuged for 5 min at 7000 g. The G-actin was polymerized to form F-actin by adding the salts KCl and MgCl2 to the final concentrations of 50 mM and 2 mM, respectively. For the batch used in the experiments, the average filament length of F-actin was detected to be approximately 7 µm under the specified polymerization condition. 26 27 2.2 Microscopy Optical microscopy is the primary technique for most of the experimental studies in this thesis. Phase contrast microscopy is used for the particle tracking microrhe- ology technique, because its enhanced visual contrast especially at the edge enables easier and more accurate particle position tracking than bright field. Fluorescence mi- croscopy allows for dynamic visualization of single actin filaments and actin bundles, since the diameter of an actin filament (8 nm) is well below the ∼ 200 nm resolution by bright field imaging. An LC Polscope system is used to measure local filament alignment and birefringence retardance value of F-actin solutions in the nematic liquid crystalline phase. The microscope used for fluorescence, phase contrast, and Polscope is a Nikon ECLIPSE E800 model. Images of 12-bit depth were collected by a 1392×1040 element Coolsnap digital camera (Roper Scientific, Trenton, NJ) driven by the MetaMorph imaging software (Molecular Devices Inc., Downingtown, PA) In this Chapter, I provide an introduction of these three microscopic techniques extensively used in this thesis study [84]. 2.2.1 Fluorescence microscopy Sir George G. Stokes discovered the phenomenon of fluorescence in 1852 when he observed that the mineral fluorophore emitted red light when it was illuminated by ultraviolet excitation, hence the name “fluorescence” [85]. Stokes also noticed that 28 the wavelength of fluorescent light is always greater than the wavelength of the exci- tation light. This effect is known as the Stokes Shift. The Stokes shift made possible the use of filters to separate the excitation light from the fluorescent emission light. In the 1930’s Haitinger and colleagues developed the technique of secondary fluores- cence, i.e. using fluorophores to stain specific tissue components, bacteria, and other pathogens that do not have the ability to autofluoresce [85]. The essential feature of any fluorescence microscope is to provide a mechanism for excitation of the specimen with selectively filtered illumination followed by the isolation of the much weaker flu- orescence emission using a second filter. This enables the formation of an image on a dark background with maximum sensitivity. Such a process is illustrated in Fig. 2.1. In a microscope the illumination and emission beams are spatially close since the microscope contains only one objective. The use of a dichroic mirror is essential. A dichroic mirror allows select wavelengths to pass through the mirror, while reflecting others. Fig. 2.2 shows how fluorescence microscopy is typically implemented in a modern microscope. The excitation and emission filters are now compressed into one cube assembly as shown in Fig. 2.2c. 2.2.2 Phase contrast microscopy The phase contrast microscopy technique was invented by Frits Zernike in the 1930s for which he received the Nobel prize in physics in 1953. Phase-contrast microscopy is a mode available on most advanced light microscopes and is most commonly used 29 Figure 2.1: Principle of excitation and emission in fluorescence microscopy. The excitation filter selects the excitation wavelength, which correlates well to the known excitation wavelength of the fluorophore. Similarly, the emission filter allows for the visualization of the weak emission light of the fluorophores at longer wavelengths. The figure is adopted from [86]. 30 Figure 2.2: Reflected light fluorescence microscopy. (a) Illumination source light (yellow light) is filtered with an excitation filter (EF). A dichroic mirror (DM) reflects the excitation beam towards the specimen. (b) The specimen is irradiated with the excitation light emitting the emission beam (red light). The emission light passes through the dichroic mirror towards the eye piece or camera port (c). The figure is adopted from [87]. to provide contrast for imaging transparent specimens such as living cells or small organisms. Transparent thin specimens are called phase objects because they slightly alter the phase of the light diffracted by the specimen. The derivation below [84] shows that the specimen retards such light approximately a quarter wavelength as compared to the undeviated direct light passing around the specimen [88]. The direct light “speeds up” by an additional quarter wavelength. Thus, the light diffracted by the specimen is retarded by a half wavelength relative to the direct light. Destructive interference between the direct and diffracted light provides the appropriate contrast. The electric field of the direct light has the form of cos(f (x, t)). The light through the sample is retarded by δ, thus 31 cos(f (x, t) + δ) = cos(f (x, t)) · cos(δ) + cos(f (x, t) − π2 ) · sin(δ) If δ is small, we can approximate to the first order π cos(f (x, t) + δ) = cos(f (x, t)) + δ · cos(f (x, t) − ) | {z } | {z 2} Direct Light Retarded Light The first term is the direct light, whereas the second term is the retarded light. Let’s introduce a phase plate that "speeds up" the direct light by a quarter wave- length. The amplitude of the propagating light becomes cos(f (x, t) − π2 ) + δ · cos(f (x, t) − π2 ) = (1 + δ) · cos(f (x, t) − π2 ) Since intensity is proportional to the amplitude squared I ∝ A2 =⇒ (1 + δ)2 ' 1 + 2δ → Linear in δ The Zernike method involves a ring annulus placed at the front focal plane of the condenser. Such an arrangement separates the illumination direct light from the diffracted light by the specimen. At the back focal plane of the objective a phase plate provides further retardation to the diffracted light, correspondingly “speeding up” the direct light (Fig. 2.3). As derived above the intensity output from such an arrangement is now linearly proportional to the specimen retardance, providing the contrast required. 32 Figure 2.3: Phase contrast microscope. A condenser annulus is inserted at the focal plane of the condenser to provide a direct light path. Conjugated to the phase plate at the rear focal plane of the objective, which "speeds up" the direct light beam by a quarter wavelength. The figure is adopted from [88]. 33 2.2.3 LC-Polscope system The LC-Polscope imaging system is a dynamic and non-destructive system for mea- suring birefringent materials. With this method, biological samples can be imaged in their native environment with no biochemical intervention for imaging. The system measures the distribution of linear birefringence using of a liquid-crystal universal compensator invented by Dr. Oldenbourg [89] and the system is manufactured by Cambridge Research and Instrumentation, Inc (Worcester, MA). The upright microscope configuration for LC-Polscope microscopy includes the insertion of a universal compensator in the illumination path prior to the condenser along with a left circular analyzer before the camera detector port. Such a configu- ration is illustrated in Fig. 2.4. The microscope condenser and objective have been specifically engineered to be strain free. The universal compensator is made of two liquid crystal variable retarders positioned at 45◦ and 0◦ with variable retardance controlled by the system. The universal compensator is controlled by a computer interface and from a set of four images the retardance magnitude and the azimuth orientation of the optical axis is computed for each pixel in the field of view, thus reporting local birefringence and filament alignment [89, 90, 91]. The algorithms for the calculation of the retardance (∆) and the optical slow axis orientation (φ) are laid out in Ref. [90]. The birefringence measurements were performed using a Nikon ECLIPSE E800 34 Figure 2.4: LC-Polscope Configuration for an upright microscope. The LC-Polscope package consists of a universal compensator inserted in the illumination path before the condenser and a left circular analyzer prior to the camera port. α and β are the retardance of the liquid-crystal plates LCA and LCB, respectively; λ/4 is a quarter- wave plate; and P and A are linear polarizer and analyzer respectively. The figure is adopted from [84]. microscope equipped with a LC-Polscope package (CRI, Worcester, MA) and a stan- dard CCD camera (MTI 300RC, Michigan, IN) with 640×480 resolution. This system produces images of the retardance and slow axis orientation of birefringent samples (for F-actin, this is the long axis of the filaments) [90]. Strain free objectives were used. 2.3 Microrheology Techniques Here I will first review briefly various microrheology techniques, and then discuss in more detail the video particle tracking (VPT) and the laser deflection particle tracking (LDPT) methods used in this thesis study. 35 2.3.1 Rheology and microrheology Rheology is the study of deformation and flow of matter under the influence of an applied stress. One of the tasks of rheology is to establish the relationship between stress and deformation. Rheology measurements are helpful in understanding struc- tural rearrangements and mechanical response of a wide range of materials, including soft materials, complex fluids and biological materials [92, 93, 94]. However, con- ventional rheology typically requires milliliter volume of sample and only measures overall bulk mechanical response of materials. To address these issues, microrheology methods emerge in order to study samples of microliters and probe local mechanical properties of materials [71, 72]. Microrheology methods extract mechanical properties of a material from the response of embedded micron-sized probes to external force or thermal excitations. Microrheology has been extensively applied to reconstituted protein systems [72, 61, 57, 56, 58, 95, 64, 63, 96, 69, 65, 62, 59, 60, 67, 66, 68] and live cells [97, 98, 99] over the last decade. Microrheology techniques mainly fall into two classes, active and passive methods. 2.3.2 Active microrheology Active microrheology requires local stress applied to probe particles by use of mag- netic field, optical trap or other micromechanical forces. Active methods often have the advantage of applying large stresses to probe stiff materials and nonequilibrium response, though they require sophisticated instrumentation in some settings. 36 In the magnetic tweezers approach (also called magnetic bead microrheometry), strong magnetic field with gradients is applied to manipulate embedded superpara- magnetic particles, and video microscopy is used to detect the displacements of the particles under the application of force [57, 56]. Three modes of operation are avail- able for magnetic tweezers approach: a viscometry measurement obtained by applying constant force, a creep response measurement after the application of a pulse-like ex- citation, or a measurement of the frequency dependent viscoelastic moduli in response to an oscillatory stress. In the optical trap approach (also called optical tweezers, or laser tweezers), a laser beam is focused onto the sample through a high numerical aperture objective lens to generate a three-dimensional trap to capture and manipulate a micron-sized dielectric particle [100]. Force is calculated by a pre-calibrated trap spring constant times the deviation of the particle from the trap center; particle position can be measured by direct imaging using video microscopy or a laser detecting technique using a quadrant photodiode detector to detect the movement of the image of laser cast through the particle [101, 58]. Then, local frequency-dependent rheological properties can be measured by oscillating the laser position with an external steerable mirror, and measuring the amplitude of the bead motion and the phase shift with respect to the driving force [101]. 37 2.3.3 Passive microrheology Passive microrheology methods extract the rheological properties of materials from the thermally excited motion of the embedded micron-sized probes without applying any external force. For passive measurements, materials must be sufficiently soft in order for the embedded particles to move detectably with only kB T of energy. In these methods, the mean squared displacement (MSD) of a single probe particle is converted to linear viscoelastic moduli via the generalized Stokes-Einstein relation us- ing various particle tracking techniques, which includes video particle tracking (VPT), laser deflection particle tracking (LDPT), and diffusing wave spectroscopy (DWS). Below I will discuss the first two methods used in my thesis study. 2.3.3.1 Video particle tracking (VPT) microrheology To implement the Video Particle Tracking (VPT) method, the motion of beads em- bedded in the material are tracked using a Nikon E800 upright microscope equipped with a Cool-Snap HQ CCD camera (Photometrics, Tucson, AZ) working in the phase contrast mode. We take time-lapse videos of beads in the middle of the capillary cavity using a 100x objective so that the beads are far away from the glass walls or any air-liquid interface. The positions of centroids of beads are obtained by us- ing the thresholding tracking algorithm in Metamorph 6.0 (Universal Imaging Corp., Downingtown, PA ). We then calculate the temporal mean square displacement (MSD) h∆r2 (t)i from 38 ˜ the bead positions. The modulus G(s) is related to the unilateral Laplace transform of h∆r2 (t)i through the generalized Stokes-Einstein (GSE) relation [71, 72]: ˜ kB T G(s) = , (2.1) πas h∆er2 (s)i where s is the Laplace frequency, kB is the Boltzmann constant, T is the absolute temperature, and a is the bead radius. This equation has been derived by Mason et al. (1997), assuming a sphere with negligible mass embedded in an incompressible and isotropically viscoelastic medium with no-slip boundary condition. The GSE relation also has a corresponding form in the Fourier domain: kB T G∗ (ω) = . (2.2) πaiωF {h∆r2 (t)i} where G∗ (ω) is the complex shear modulus, ω is the angular frequency, i is the imaginary unit, and F represents Fourier transformation. Since the data of h∆r2 (t)i are in discrete times and over a limited range, im- plementing a numerical Laplace transform introduces significant truncation errors near the frequency extremes, whereas it does give accurate results well within the region between the two extremes. Similar errors near the frequency extremes may also occur when the Fast Fourier Transform is used. To overcome the errors, Ma- son et al. [102] estimated the transforms algebraically by applying a local power law expansion around a frequency of interest ω, and retaining the leading term, h∆r2 (t)i = h∆r2 (1/ω)i (ωt)α(ω) . The evaluation of the Fourier transform can be given as 39 ­ ® ­ ® iωF( ∆r2 (t) ) = ∆r2 (1/ω) [Γ(1 + α(ω))] i−α(ω) . (2.3) ¯ d lnh∆r2 (t)i ¯ where α is defined as α(ω) ≡ d ln t ¯ . Substituting Eq. 2.3 into Eq. 2.2 leads t=1/ω to G0 (ω) = |G∗ (ω)| cos(πα(ω)/2) , (2.4) G00 (ω) = |G∗ (ω)| sin(πα(ω)/2) , (2.5) where G0 (ω) is the storage modulus and G00 (ω) is the loss modulus, and |G∗ (ω)| = kB T / {πa h∆r2 (1/ω)i [Γ(1 + α(ω))]}. Eq. 2.4 and Eq. 2.5 yield a helpful physical interpretation of moduli in terms of MSD. For a viscous medium, diffusion dominates and α approaches one; G00 becomes prominent and G0 vanishes. For a more elastic medium, the motion is more confined and α approaches zero; G0 dominates and G00 diminishes. In Chapter 4, we extend this microrheology technique to F-actin solutions in the nematic phase. Assuming that the motion of test spheres in the direction parallel to the nematic director is only determined by G∗|| (ω) and the motion in the perpendicular direction by G∗⊥ (ω), we extend the analytical treatment summarized above to an anisotropic medium to obtain G0|| (ω), G00|| (ω), G0⊥ (ω) and G00⊥ (ω) from two independent kBDT 1-D diffusions. Note that Eq. 2.2 needs to be modified to G∗|| (ω) = E 2 (t) ) 2πaiωF ( ∆r|| 40 kB T and G∗⊥ (ω) = 2 (t) ) , 2πaiωF (h∆r⊥ respectively, where the factor of 1/2 arises from the i fact that the MSD is now 1-D for either direction instead of 2-D. 2.3.3.2 Laser deflection particle tracking (LDPT) microrheology The experimental setup for the LDPT method is similar to the one described pre- viously in Ref. [58] (Fig. 2.5). The illuminating laser beam generated by a diode pumped Nd:YAG laser source at 1064 nm infrared wavelength (CrystaLaser LC, Reno, NV) is steered into a Nikon Eclipse TE 2000-U inverted microscope with an external optical train, and is introduced from the epi-fluorescence port and deflected into the optical path of the microscope by a dichroic mirror located below the objective lens. Laser is focused on the sample to form an optical trap. The laser power is tuned down by orders of magnitude compared with that required to produce a laser trap, but it is still intense enough for accurate position detection. Particle position detection is implemented by detecting the position of the shadow cast by the particle onto a quadrant photodiode detector (custom made by Mr. Winfield Hill, Roland Institute, Cambridge, MA). Voltage outputs from the photodiode detector are acquired by a BNC 2090 board and processed by LabVIEW (National Instruments, Austin, TX). When the particle is in the center of the trap, the intensity of light (in voltage signal) on each of the four quadrants is equal. As the particle moves from the trap center, the difference in light intensity on the four quadrants is recorded to measure the particle displacement: the total intensity of the top two quadrants minus that of the bottom 41 Figure 2.5: An outline of the essential components of the laser and optical paths. The figure is adopted and modified from [58]. two is proportional to the X displacement within the linear range with a prefactor to be determined, and we can similarly obtain the Y displacement by subtracting the right total intensity from the left. With the known thermal motion of particle in water, we can determine the calibration prefactor, and hence, obtain the positional information of particle motion [58]. To convert the positions of beads to moduli, we follow the procedure described by Schnurr et al. [63] & Gittes et al. [64], using the generalized Stokes (GS) rela- tion. When a force is applied on a sphere, the linear response for the sphere can be calculated from the deformation of the surrounding medium, rω = α(ω)fω where r is the position of the sphere. Generally, the response function has a complex form 42 α∗ (ω) = α0 (ω) + iα00 (ω). This response function is related to the complex shear modulus by the GSR 1 α∗ (ω) = . (2.6) 6πG∗ (ω)a This equation has been derived and justified in Ref. [63]. The imaginary part of the response function α00 (ω) can be related to the power spectral density (PSD) of 1-D D E 2 thermal motion r||,⊥ (ω) (Fourier transform of MSD) of either parallel or perpen- dicular direction by the fluctuation-dissipation theorem [103] as 00 ω ­ 2 ® α (ω) = r||,⊥ (ω) . (2.7) 2kB T 00 The real part α0 (ω) is obtained from the imaginary part α (ω) via the Kramers-Kronig relation [103] by evaluating the dispersion integral Z∞ Z∞ Z∞ 0 2 ζα00 (ω) 2 α (ω) = P dζ 2 2 = dt cos ωt dζα00 (ζ) sin(ζt) , (2.8) π ζ −ω π 0 0 0 where P indicates the principal-value integral. Finally, G∗ (ω) is obtained from the reciprocal of α∗ (ω), 0 α0 (ω) G (ω) = , (2.9) 6πa(α02 (ω) + α002 (ω)) −α00 (ω) G00 (ω) = . (2.10) 6πa(α02 (ω) + α002 (ω)) For a nematic solution, the response function in the parallel and perpendicular ∗ directions α||∗ and α⊥ are different. Assuming decoupling of the motion in the two directions, we consider separate response functions, r|| (ω) = α||∗ (ω)f|| (ω) and r⊥ (ω) = ∗ α⊥ (ω)f⊥ (ω). Using Eqs. 2.7 to 2.10, we obtain accordingly G0|| (ω), G00|| (ω), G0⊥ (ω) and G00⊥ (ω) . 43 2.3.3.3 VPT vs. LDPT methods The VPT and the LDPT methods each have their own advantages and limitations. The VPT method can track multiple particles at the same time and is convenient to implement, yet it is not capable of giving high frequency information due to the limited resolution and response time of the CCD camera. Additionally, since the amount of memory available to take the video is limited, the VPT method cannot take more than a few thousand frames. Consequently, the direct Fourier transform introduces a significant truncation error across the whole frequency range, so the power expansion method is more suitable for the VPT method. On the other hand, the laser has much faster response time to changes of the bead position in order to obtain the high frequency spectrum. The LDPT method readily acquires data of millions of positions, which can be efficiently processed by the direct Fourier transform but not by the power expansion method. This is because it takes a long time for the power expansion method to compute the slope and moduli at each point. Because of the large amount of data, leaving out the regions at both low and high frequency extremes with large truncation errors still yields accurate results in a wide enough range. The disadvantage of the LDPT method is that the focused laser beam applies a trapping force on the tracer bead and alters its free motion; it requires a calibration of the conversion factor from voltage to displacement every time and needs to have the trap stiffness subtracted from G0 . For measuring the dependence of moduli on actin concentration and ion condition, we are mostly interested in the low frequency range, 44 where the VPT method yields reliable results. Therefore, we use primarily the VPT method for the measurements of shear moduli. LDPT, as a complementary approach, is performed under selected conditions to show the high frequency behavior. 2.3.4 Two particle Microrheology Single particle method probes the local properties of materials in the length scale of the probe radius, and is sensitive to probe size, probe surface properties, and local inhomogeneity [104]. To address these issues, two particle method was developed, which tracks the cross-correlated motion of a pair of probe particles to obtain the long wavelength viscoelastic properties of the embedded material [96, 69, 66]. Two particle method probes the properties of materials in the length scale longer than the distance between the particles; it is insensitive to the size of the tracer particles, material inhomogeneity and specific coupling between probe and the medium [66, 68]. Two particle method is advantageous to be consistent more with bulk rheology measurements, yet, it can not probe short length scale properties which are detectable by single particle method. In the two particle method, the particle position detection is typically implemented through video tracking, where hundreds of tracer particles can be tracked simultane- ously [69]. Vector displacements of individual tracers are calculated as a function of lag time τ , and absolute time t as ∆r(t, τ ) = rα (t + τ ) − rα (t). Then, the ensemble averaged tensor product of the vector displacements is calculated, 45 ­ ® Dαβ (r, τ ) = rαi (t, τ )rβj (t, τ )δ(rij − R(t) i6=j,t , (2.11) where i and j are the labels of two particles, α and β are coordinate axes, and Ri,j is the distance between particles i and j. The average is taken over only the distinct terms (i 6= j); the “self” term yields the one-particle MSD, h∆r2 (τ )i. For an incompressible medium, the expected two point correlation is computed by multiplying the displacement predicted in Eq. (2.1) by the strain field of a point stress [105]. In the limit of r À a, the result is ˜ rr (r, s) = kB T 1 D , Dθθ = Dφφ = Drr , (2.12) ˜ 2πrsG(s) 2 ˜ rr (r, τ ) is the Laplace transform of Drr (r, τ ) and the off-diagonal tensor el- where D ements vanish. Noting that Eq. (2.12) has no dependence on a, suggesting that Drr (r, τ ) is independent of the tracer’s size, shape and boundary conditions with the medium when r À a . Comparing the longitudinal two-point correlation (Eq. (2.12)) and the GSE equa- tion (Eq. (2.1)) suggests that we define a distinct MSD, h∆r2 (τ )iD as ­ ® 2r ∆r2 (τ ) D = · Drr (r, τ ) , (2.13) a This quantity is just the thermal motion obtained by extrapolating the long-wavelength thermal undulations of the medium down to the bead size. h∆r2 (τ )i may be under- stood as a superposition of a long-wavelength motion described by h∆r2 (τ )iD plus a 46 local motion in a cavity. To obtain two particle microrheology results, the distinct MSDs h∆r2 (τ )iD are substituted into the GSE equation in place of h∆r2 (τ )i. Chapter 3 Surface Adsorption Causes an Opposite Probe Size Dependence to that of Hopping in Microrheology of Actin Network 3.1 Introduction Despite the amazing diversity in size, shape and function of many eukaryotic cell types, all of them consist of a prominent cytoskeleton formed from a network of protein filaments [4]. Understanding the mechanical properties of the cytoskeleton is crucial for assessing biological functions as diverse as cell motility, wound healing, 47 48 organelle transport and phagocytosis. The predominant component of this network is the actin filament. With a diameter of only 8 nm [23], the polymerized actin filaments are thin, but stiff, with a persistence length on the order of 10 micrometers [27, 28]. These rodlike filaments are of variable lengths and form highly compliant and deformable networks that are dynamically regulated in live cells, thereby performing their essential biological functions. Because of the singular role of actin networks in cell mechanics, there has been extensive investigation into the rheological properties of actin networks reconstituted from pure actin [92, 94, 62, 106, 66, 73, 2], in some cases with accessory actin binding proteins including those that crosslink actin filaments [107, 108], induce new branches [106], or cap filament ends to terminate their growth [109]. The results of these studies with reconstituted protein mixtures have reproduced in many aspects the viscoelasticity of live cells, and unraveled many intriguing properties of cells that are mechanical in nature. For example, the typical elastic modulus of mammalian cells, on the order of 103 Pa [110], can be produced by crosslinked actin networks of as little as 1% protein content [93]. Additionally, the major features of a pre-stressed state, typical of live cells, have been mimicked by adding small aggregates of the actin motor protein myosin into the actin network [99]. Over the past decade or so, particle tracking microrheology has been demonstrated as the most practical method for probing cell mechanics [72, 61, 95, 96, 69, 97, 98, 99]. Using micron- or submicron-sized probe particles, one extracts the mechanical 49 properties of materials by measuring thermally driven motion [72, 63, 64, 61, 60, 59, 62], or motion driven by an external force such as that generated by laser [101, 58, 111] or magnetic tweezers [57, 56]. Since the probe size can be significantly smaller than the size of typical cells, local mechanical properties can be probed with sub-cellular resolution. Under certain situations, endogenous particles or even spots of labeled proteins naturally integrated into the networks of interest can be used as probes to acquire the viscoelastic properties of their immediate environment [112]. It has been increasingly recognized that many factors complicate the interpretation of microrheology data. The most obvious limitation of single particle tracking is that probe particles unavoidably interact with the surrounding medium, thus reporting mechanical properties that have been altered by their presence. One method has been shown to essentially overcome this local effect, by measuring cross-correlations between two probe particles with a distance between them much larger than their own size [69, 96, 66, 68]. This two particle method probes the properties of materials in the length scale longer than the distance between the particles, and is insensitive to short length scale properties and specific coupling between the probe and the medium [66, 68]. The two particle microrheology measurements are shown to be more consistent with the bulk rheology results, though it ignores the local microscopic properties around each bead. The actual cellular environment, however, is rich in structure and often renders the cross-correlation technique inapplicable. Therefore, single particle method remains advantageous in probing cell mechanics, since it is 50 capable of reporting local properties in short length scale of the probe radius [104], which is usually not accessible by the two particle method. In this Chapter, we address two commonly occurring factors that affect single particle microrheology: surface adsorption and hopping, both of which cause the measured rheological properties to be inaccurate and dependent on the probe size. Surface adsorption reduces the motion of micron sized probe beads, resulting in an artificial increase in measured stiffness [65, 67]. Hopping is the process by which small adsorption resistant probe particles hop over cages defined by the mesh size of the surrounding network. This hopping effect has been reported previously [66, 73]. Here we report new measurements to systematically assess both effects on actin microrheology by varying the surface coating (adsorption) and probe size. Through this study we find that these two effects cause opposite trends in the probe size dependence of network stiffness. Therefore, certain undesirable effects, such as a biased estimate of the shear moduli and probe size dependency, may be eliminated by proper treatment of the probe beads to make the two opposite effect compensate for each other. We also confirm the existence of a depletion layer around non-sticky probe surface through a confocal image analysis assay, and qualitatively analyze its effect on probe size dependency. Our work provides a simple guideline for optimization of probe surface chemistry and precise interpretation of microrheology data. 51 3.2 Materials and Methods 3.2.1 Preparation of BSA coated PS beads Rhodamine labeled bovine serum albumin (BSA) in dry powder form (Invitrogen Corporation, Carlsbad, CA) was suspended in deionized water at a concentration of 2 mg/mL. Carboxylate polystyrene (PS) beads (Polysciences, Inc., Warrington, PA) with diameters of 0.45, 1.0, 2.0, 3.0 and 4.5 µm were added to Rhodamine BSA solutions in separate tubes and incubated at room temperature for 1 hour. A 20 mg/mL BSA stock solution was then added in excessive volume and incubated for half an hour to further block open adsorption sites. The beads were then centrifuged at 6000 rpm for 5 minutes and the supernatants were discarded. Beads were resuspended with deionized water. The mixed suspension was centrifuged to retain beads in the pellet. This wash process was repeated two more times. Prepared beads were kept in deionized water at 4 ◦ C and used within one week after preparation. The same beads were used for both microrheology measurements and confocal imaging. 3.2.2 Preparation of PEG grafted PS beads To obtain the PEG coated PS beads, we followed the standard carbodiimide cou- pling chemistry used by Valentine et al. [67] to attach amine-terminated methoxy- poly(ethylene glycol), NH2 -(CH2 -CH2 -O)n -OCH3 (mPEG-NH2 ), with average n=16 52 and average molecular mass of 750 Da (Rapp Polymere, Tübingen, Germany), to car- boxylate PS microspheres. EDC (1-[3-(dimethylamino)propyl]-3-ethylcarbodiimide) (Sigma-Aldrich, St. Louis, MO) reacts with a carboxyl group to form an amine- reactive O-acylisourea intermediate, which is an unstable ester and will hydrolyze and regenerate the carboxyl group. With the presence of NHS (N-hydroxysuccinimide) at a lower pH, EDC converts carboxyl groups to the more stable amine-reactive NHS esters. The activated esters are then mixed with mPEG-NH2 at a higher pH to react with the NHS-ester to yield a stable amide bond. To avoid clustering of particles due to traditional centrifugation or filtration, we followed a modified buffer exchange protocol using dialysis tubes as described in Ref. [67]. Microspheres (1 mL for each size) were loaded into dialysis tubes (SpectraPor, 10 kD cutoff; Spectrum, Rancho Dominguez, CA) at number densities of 1011 –1013 particles/mL; higher number densities result in aggregation and poor coupling effi- ciency. The bags were immersed in MES (100 mM 2-(N-morpholino)ethanesulfonic acid) buffer (100 mM MES at pH 6.0) for 2 hours to adjust the pH. All the dialysis processes were performed under constant gentle stirring. Then, the bags were sub- merged into MES buffer containing 75 mM EDC, 50 mM NHS, and a 200-fold excess of mPEG-NH2 to carboxylate group for 1 hour to activate the reaction. The bags were then immersed into the borate reacting buffer (50 mM boric acid, 36 mM sodium tetraborate, 50 mM NHS, and 200 fold excess of mPEG-NH2 , pH 8.5) to allow reac- tion to proceed for 8 hours. This step was repeated twice. After the second reaction, 53 the particles were washed with pure borate buffer (50 mM boric acid, 36 mM sodium tetraborate, pH 8.5) twice for 2 hours to remove any unreacted reagents and polymer. Finally, the particles were recovered and stored in the pure borate buffer at 4 ◦ C, and can be used over several months. 3.2.3 Sample preparation for microrheology Probe beads used in the microrheology measurements were carboxylate PS beads (Polysciences Inc., Warrington, PA), silica beads (Bangs Laboratories, Inc., Fishers, IN), BSA coated PS beads and PEG coated PS beads, prepared as described above. Diameters of the PS beads were 0.45, 1.0, 2.0, 3.0 and 4.5 µm; diameters of the silica beads were 0.32, 0.57, 1.0, 2.0, 3.0 and 4.5 µm. Beads were diluted with actin buffer and then mixed with a polymerized actin solution and allowed sufficient time to be incorporated and equilibrated into the actin network. The sample was then injected into glass-capillaries with cross-sectional dimensions of 0.1 mm × 1 mm (VitroCom Inc., Mt. Lakes, NJ) for convenient microscopic observation. The filled capillary was sealed with an inert glue to eliminate flow and evaporation. 3.2.4 Preparation of Alexa-488 labeled actin for confocal imag- ing Fluorescently labeled G-actin used for confocal imaging was prepared following the procedure in Ref. [75]. G-actin was labeled on a random amino acid group using 54 Alexa 488-succinimidyl ester (Invitrogen Corporation, Carlsbad, CA). G-actin was dialyzed against P-buffer (50 mM PIPES, pH 6.8, with 50 mM KCl, 0.2 mM CaCl2 , and 0.2 mM ATP) to polymerize while removing the interfering Tris present in the actin buffer. Alexa 488-succinimidyl ester (30 mM), dissolved in dimethyl formamide, was added drop-wise to a 7.5-fold molar excess over actin (typically 60 µM). After incubation for one hour at room temperature in the dark, the reaction was quenched by adding 20 mM Tris, pH 8.0, 16 mM ATP, 10 mM K-glutamate, 5 mM lysine, and 5 mM DTT. Followed by dialysis against the buffer (2 mM Tris, pH 8.0, with 0.2 mM ATP, 0.5 mM DTT, 0.25 mM CaCl2 , 0.05 mM EDTA, and 0.05% sodium azide), the Alexa 488 labeled actin was stored frozen in 10% sucrose (w/v) at -80°C. Using a spectrophotometer to determine the labeling efficiency with manufacturer’s extinction coefficients, labeling result was found to be substoichiometric (typically 0.42-0.5 in molar ratio). 3.2.5 Confocal imaging of beads and actin network The visualization of the actin network and Rhodamine BSA coated bead surface was performed with a Leica confocal Laser scanning microscope, using 100x oil immersion lens. The images taken were in 512 × 512 pixels, covering a 49.8 × 49.8 µm region. Alexa-488 labeled actin was excited by an argon laser (wavelength 488 nm) and the Rhodamine BSA was excited by a helium-neon green laser (wavelength 543 nm). The samples for confocal imaging were filled between a coverslip and glass slide, sealed 55 a) Carbox PS b) Silica c) Rhod BSA PS d) PEG PS 5 µm Figure 3.1: Confocal images of 0.34 mg/ml Alexa-488 labeled F-actin solutions em- bedded with different beads visualized at the bead equator section. Four types of beads are shown: (a) carboxylate PS beads, (b) silica beads, (c) Rhodamine BSA coated PS beads, (d) PEG coated PS beads, all with 1 µm beads shown in the top row, and 4.5 µm beads shown in the bottom row. The white arrows indicate the locations of the 1 µm beads. A scale bar of 5 µm, shown at the lower right corner, applies to all the images. with vacuum grease. The sample thickness was between 20 and 25 µm. The images of beads and actin network were acquired at the equator of the beads using corresponding excitation lasers, while keeping the field of view fixed. The images shown in Fig. 3.1 were cropped regions of 130 × 130 pixels from the original images. 56 3.3 Results 3.3.1 Direct comparison of protein adsorption by confocal im- ages Protein filaments such as F-actin tend to physisorb on the surface of many types of probe beads [65, 67]. Before addressing this surface adsorption effect on microrheol- ogy, we first compared fluorescently labeled actin filament networks embedded with the four types of beads using a confocal microscope. Figure 3.1 compares represen- tative images of both 1 µm and 4.5 µm beads of the four types of beads, namely, (a) carboxylate PS, (b) silica, (c) PEG coated PS, and (d) Rhodamine BSA coated PS, all embedded within the 0.34 mg/ml actin network. Rhodamine labeled BSA was used to coat the PS beads, so that the surface of the BSA beads could be determined by the red fluorescence profile, to be shown later with the intensity profile analysis. The green fluorescence on the surfaces of carboxylate PS beads and silica beads is due to the adsorbed actin filaments, with the former showing brighter and thicker rings indicative of stronger adsorption. There is almost no adsorption on the BSA and PEG beads, and their differences in actin coating, if any, is hard to detect by confocal imaging. This depletion effect is well known to colloidal scientists. Most re- cently, such an effect has been shown and analyzed in detail between actin networks and a flat microscope slide surface [75]. Later, we will analyze the depletion effect of spherical particles quantitatively using a similar approach. 57 3.3.2 Physisorption leads to size dependent viscoelasticity mea- surement In order to assess the effect of surface adsorption on the detection of network me- chanics, we performed particle tracking measurements using all four types of beads, thus varying the level of surface stickiness. For carboxylate PS beads, silica beads, and BSA coated carboxylate PS beads, we note that the average motion of the probe beads increased in the order listed, corresponding to that of decreasing stickiness. For each surface property among these three bead types, smaller beads displayed larger MSDs, as represented by BSA beads in Fig. 3.2a. The GSE relation predicts the MSD to be inversely proportional to the diameter of probe beads so that the mea- sured moduli are independent of probe size. When the measured MSD was scaled by multiplying the diameter of probe beads, however, our data show a residual de- pendence on the probe size, with the scaled MSD increasing slightly with bead size, as shown in Fig. 3.2b. After the MSD data were Fourier transformed to yield the storage and loss moduli G0 and G00 through the GSE relation, both values appeared to be higher when probed by smaller beads over the entire frequency range, as shown in Fig. 3.3. Therefore, for a probe with sticky surface, smaller beads report larger stiffness of the network. Physisorption affects the GSE relation and produces the probe size dependence, but not drastically. In the extreme case when the probe beads are tightly attached to the network, the measured MSD would be insensitive to the bead size, thus the scaled 58 −1 10 a) MSD (µm ) 2 −2 10 0.45 µm 1 µm 2 µm 3.1 µm 4.5 µm −3 10 −1 0 1 10 10 10 t (sec) −1 10 b) MSD*a (µm ) 3 −2 10 0.45 µm 1 µm 2 µm 3.1 µm 4.5 µm −3 10 −1 0 1 10 10 10 t (sec) Figure 3.2: MSDs and scaled MSDs of BSA coated PS beads in a 0.34 mg/mL F- actin solution. (a) The MSDs as functions of time interval are plotted for a series of PS beads coated with BSA. All the beads with different sizes are measured in the same F-actin solution to eliminate sample-sample variation. (b) The scaled MSDs (MSD×a) of BSA coated PS beads as functions of time interval are obtained from the MSD data in (a). 59 a) Storage Modulus (Pa) −1 10 G’ 0.45 µm −2 G’ 1 µm 10 G’ 2 µm G’ 3 µm G’ 4.5 µm −1 0 10 10 ω (rad/sec) b) −1 10 Loss Modulus (Pa) G’’ 0.45 µm −2 G’’ 1 µm 10 G’’ 2 µm G’’ 3 µm G’’ 4.5 µm −1 0 10 10 ω (rad/sec) Figure 3.3: The frequency spectra of storage (a) and loss (b) moduli of a 0.34 mg/ml F-actin solution measured by different sized BSA coated PS beads obtained from the MSD data shown in Fig. 3.2. 60 MSD would be proportionally smaller for the smaller beads, leading to significantly larger G0 and G00 values. In our experiments, however, even the most-sticky carboxy- late PS beads did not approach this extreme behavior. Our measurements yielded a smaller MSD for the PS beads without coating, followed by the silica beads and the BSA coated PS beads. This order is consistent with their decreasing stickiness as anticipated by their distinct surface chemistry. 3.3.3 Cage hopping detected using adsorption resistant beads smaller than the network mesh size The PEG functionalized PS beads fall into the class of adsorption resistant probes. They display more motion in the actin network compared with three other sticky probes, consistent with the known “slippery” nature of these beads. At the low protein concentration of 0.34 mg/ml actin, which forms a network with a mesh size about 0.51 µm [113], the PEG-beads of 0.45 µm diameter moved through the network transiently, resulting in an MSD much larger than expected for those trapped in a protein network (Fig. 3.4a). The larger beads were progressively more constrained by the surrounding actin network. As a result, the scaled MSD decreased with the bead diameter (Fig. 3.4b). This opposite trend in scaled MSD translated into an opposite dependence of shear moduli on bead size, as well. As shown in Fig. 3.5, both G0 and G00 increased with the bead diameter across the entire frequency range when probed by PEG coated beads, an opposite behavior to the other three types of 61 a) 0 MSD (µm ) 10 2 −1 10 −2 10 0.45 µm 1 µm 2 µm −3 10 3 µm 4.5 µm −1 0 1 10 10 10 t (sec) 1 10 b) 0 10 MSD*a (µm ) 3 −1 10 0.45 µm −2 10 1 µm 2 µm 3 µm 4.5 µm −3 10 −1 0 1 10 10 10 t (sec) Figure 3.4: MSDs and scaled MSDs of PEG coated PS beads in the 0.34 mg/mL F- actin solution. (a) The MSDs of PEG coated PS beads as a function of time interval are plotted for five different bead diameters, including 0.45, 1.0, 2.0, 3.0 and 4.5 µm. The MSDs are measured when all the beads with different sizes are embedded in the same F-actin solution to eliminate sample-sample variation. Each MSD curve is an average of 10 beads. (b) The scaled MSDs (MSD×a) of PEG coated PS beads in the 0.34 mg/mL F-actin solution are obtained from the MSD data in (a). 62 a) Storage Modulus (Pa) −2 10 G’ 0.45 µm G’ 1 µm −3 10 G’ 2 µm G’ 3 µm G’ 4.5 µm −1 0 10 10 ω (rad/sec) −1 10 b) Loss Modulus (Pa) −2 10 G’’ 0.45 µm −3 10 G’’ 1 µm G’’ 2 µm G’’ 3 µm G’’ 4.5 µm −1 0 10 10 ω (rad/sec) Figure 3.5: The frequency spectra of storage (a) and loss (b) moduli of 0.34 mg/ml F-actin solution measured by different sized PEG coated PS beads obtained from the MSD data shown in Fig. 3.4. 63 1 0.34 mg/ml PEG bead 0.34 mg/ml BSA bead 0.34 mg/ml silica bead 0.34 mg/ml carbox bead m ) 1.0 mg/ml PEG bead 1.0 mg/ml BSA bead 2 1.0 mg/ml silica bead 0.1 1.0 mg/ml carbox bead Scaled MSD at 1 sec ( 0.01 1E-3 1 Bead Diameter ( m) Figure 3.6: For beads of four different surface chemistries, the scaled MSDs at 1 second as functions of bead diameter are plotted for actin concentrations of 0.34 (open symbols) and 1.0 mg/mL (solid symbols) in a log-log plot. The PEG PS bead data are shown in green squares, the BSA PS bead data in red circles, the silica bead data in blue triangles, and the carboxylate PS bead data in black inverted triangles. beads. Note that the 0.45 µm PEG beads detect significantly smaller moduli, for they percolate through the network. Thus, for the adsorption resistant slippery probes, smaller beads probe a much softer network. 64 3.3.4 Opposite trends of scaled MSD vs. probe size for ad- sorption and non-adsorption surface chemistries To demonstrate the opposite trends more clearly, we plot the scaled MSD at 1 second as a function of bead diameter for all four surface chemistries at two actin concen- trations of 0.34 and 1.0 mg/mL (Fig. 3.6). The magnitude of the scaled MSDs of different bead types are consistent with their extent of surface adsorption. In the order of decreasing surface stickiness, carboxylate PS beads, silica beads, BSA coated PS beads, and PEG coated PS beads have increasing magnitude of the scaled MSD for both actin concentrations, with the separation among different bead types more striking for the more dilute actin solution. The slope (or scaling exponent) of the scaled MSD vs. bead diameter in a log-log plot correlates with the probe surface properties. In the ideal case, the scaled MSD is independent of probe size, and the slope should be zero. For the three types of beads with protein adsorption, smaller beads have smaller scaled MSDs. Thus, their plots in Fig. 3.6 have positive slopes. Also, stickier beads show steeper positive slopes. This feature is more striking for the dilute 0.34 mg/mL solution; for the 1 mg/mL solution, the difference in the scaled MSD between the three types of beads tends to diminish. For the adsorption resistant slippery beads, smaller PEG beads show large scaled MSDs, resulting in a negative slope. For the largest bead size of 4.5 µm, all bead types tend to converge at the higher actin concentration of 1 mg/mL. 65 3.3.5 Opposite trends of moduli vs. probe size for adsorption and non-adsorption probes We now compare the dependence of storage modulus G0 and loss modulus G00 on the probe size for 0.34 and 1.0 mg/mL actin solutions measured by all four types of beads (Fig. 3.7). The comparison shows the opposite trends of probe size dependence of moduli measured by the adsorption and non-adsorption beads. G0 and G00 have similar features. For both G0 and G00 , carboxylate beads detect the largest values, followed in the descending order by silica beads, BSA PS beads, and PEG beads. Both G0 and G00 depend on probe size for all four bead surface properties, though for PEG beads G0 has a very weak size dependence with the exception of the smallest bead size. Since shear moduli are proportional to the inverse of Fourier transformation of scaled MSD (Eq. 2.2), one expects that the trend of measured moduli vs. probe diameter is reversed from the scaled MSDs. For the slippery PEG beads, the smaller beads detect smaller moduli, hence, a positive slope is observed for moduli vs. bead diameter. For the other three types of beads, the smaller beads detect larger moduli, so we observe a negative slope. Table 3.1 summarizes the trends of the scaled MSDs and shear moduli measured by four different types of beads. Consistent with the feature for scaled MSDs, the reported shear moduli among all four types of beads tended to converge with increasing bead size and actin concentration. The reason for this trend is that the surface effect becomes less important when either parameter increases, but the detailed mechanisms may differ and will be later discussed in more 66 a) 1 0.1 G'(Pa) 1.0 mg/ml carbox 0.01 1.0 mg/ml Silica 1.0 mg/ml BSA PS 1.0 mg/mL PEG PS 0.34 mg/ml carbox 0.34 mg/ml Silica 0.34 mg/ml BSA PS 0.34 mg/ml PEG PS 1E-3 1 Bead Diameter ( m) b) 0.1 G''(Pa) 1.0 mg/ml carbox 1.0 mg/ml Silica 0.01 1.0 mg/ml BSA PS 1.0 mg/mL PEG PS 0.34 mg/ml carbox 0.34 mg/ml Silica 0.34 mg/ml BSA PS 0.34 mg/ml PEG PS 1 Bead Diameter ( m) Figure 3.7: The storage (a, solid symbols) and loss (b, open symbols) moduli as functions of bead diameter are shown for two actin concentrations, 0.34 mg/mL (dash lines) and 1.0 mg/mL (solid lines). Carboxylate PS beads are in black inverted triangles, silica beads in blue triangles, BSA PS beads in red circles, and BSA PS beads in green squares. The storage and loss moduli were adopted at 1 Hz from the modulus spectra converted from MSD vs. time data. 67 bead type stickiness scaled MSD slope of scaled MSD slope of moduli smaller bead senses PEG PS - 1st (largest) - + softer network BSA PS + 2nd + - more rigid network silica ++ 3rd ++ -- more rigid network carbox PS +++ 4th (smallest) +++ --- more rigid network Table 3.1: Correlation of the probe size dependence of microrheology with probe sur- face stickiness. For the stickiness, more “+”s indicate more sticky; more “-”s indicate more slippery. For the slopes, more “+”s indicate a larger positive value; more “-”s indicate a negative number with a larger absolute value. detail. 3.3.6 Depletion effect detected from BSA coated beads To assess the depletion effect of adsorption resistant or weak adsorption probes, we analyzed the radially averaged intensity profile of Rhodamine labeled BSA and the Alexa-488 labeled actin in the confocal images of 4.5 µm BSA coated beads (Fig. 3.8). BSA coated beads are selected since their surface boundary can be conveniently determined from the Rhodamine BSA fluorescence profile. The position of the surface boundary is at the radial distance 2.23 µm from the bead center, determined by the peak position of BSA intensity profile. The bead diameter is calculated as 2.23 × 2 = 4.46 µm, consistent with the manufacturer supplied bead diameter of 4.452 µm. The location of 50% maximum intensity of the actin fluorescence is at about 2.52 µm from the bead center. Fisher and Kuo empirically defined the distance between this location and the surface to be the depletion layer thickness [75]. According to this definition, we detect for the 0.34 mg/mL actin network a depletion layer thickness of 0.29 µm. However, because of the spherical shape of probe particles and the limited 68 a) 5 µm b) 160 140 radially averaged intensity 120 100 80 60 40 20 layer 50% of Max thickness 0 0 1 2 3 4 5 6 7 8 9 radial distance (µm) Figure 3.8: (a) The confocal image of labeled F-actin solution (left) embedded with the 4.5 µm BSA coated bead is re-shown with the confocal image of Rhodamine BSA coated on the bead surface (right). The dashed circles are obtained from the center threads of the bright fluorescence rings, which define the boundaries of the BSA coated beads. (b) Radially averaged intensity profile from the center of the bead shown in (a), with the red solid line showing the intensity profile of Rhodamine labeled BSA and the green dash line showing the profile of Alexa-488 labeled actin. The radial distance is originally in the unit of pixel and converted to µm with a calibration factor 0.097 µm per pixel. The radially averaged intensity profiles are obtained by drawing circles with radial increment of 1 pixel each time starting from the center of the bead, and dividing the sum of all the pixel intensities between consecutive circles by the total number of pixels counted. The vertical dash dot line shows the peak position of BSA intensity profile. The blue dot shows the 50% of maximum intensity of actin profile. The distance in between is the depletion layer thickness defined by Fisher and Kuo [75]. 69 z-resolution of the confocal microscope used, the fluorescence on the curved surface within the outmost boundaries is also collected, which is expected to shift the 50% maximum intensity position towards the center of the bead. Thus, this 0.29 µm is the lower limit of the depletion layer thickness. With this complicating factor in mind, the depletion layer thickness we detect is consistent with the depletion layer from a BSA coated flat surface of about 0.5 µm, as obtained by Fisher and Kuo [75]. Therefore, we confirm that the depletion effect clearly exists around beads with a weak protein adsorption surface, the extent to which it affects the microrheological data is to be discussed below. 3.4 Discussion Particle tracking microrheology is an effective tool to probe the micromechanical properties of polymeric materials, biomaterials and cells. However, complications like probe size dependence, probe surface chemistry, depletion effect and sample inhomo- geneity impair the precise interpretation of microrheological data. Despite previous efforts by others who have characterized the effects of probe surface chemistry on particle tracking microrheology [65, 67], our new study yields several advances in the field. First, through a systematic comparison of actin networks embedded with four types of bead surfaces, each type with five different diameters, we were able to clearly identify the opposite effects in terms of size dependence of scaled MSDs be- tween “sticky” and “slippery” beads. Second, we proposed distinct effects of surface 70 H opp ing D epletion L a ye r S urface A dsorption Figure 3.9: A schematic illustration of three probe beads embedded in a network of actin filaments. Surface adsorption is illustrated for one sticky bead shown in gray. Also depicted are two slippery beads, shown in blue, with the smaller one hopping through the porous network and the larger one trapped in the same network. There is a notable depletion in filament density near the surface of the larger bead. adsorption and hopping, which account for and are consistent with the opposite size dependence in the values of shear moduli. The phenomena of surface adsorption, de- pletion and particle hopping are illustrated in Fig. 3.9. Finally, we have made direct visualizations of filament adsorption on the selected types of probe bead surfaces, as well as the depletion effect for the cases where probe beads are “slippery”, i.e., non-sticky. Our assessment of surface “stickiness” based on microrheology data proves to be consistent with direct imaging using fluorescent tags. In certain cases the former is even more sensitive, evidenced by the difference it shows between the BSA and PEG 71 beads. The consistency of both techniques also suggests the usefulness of microrheol- ogy for determining probe-network interactions under situations where direct imaging might not be feasible, or doing so might adversely alter the network properties. An additional advantage of the microrheology assay is that it is not limited by the optical diffraction limit of co-localization. Instead, the centroid of a micron-sized bead can be determined to nanometer resolution [2]. The sticky beads binding to surrounding filaments result in a negative slope of the measured moduli vs. probe diameter, i.e. smaller beads detect stiffer networks. When filaments stick to the bead surface, they act like semiflexible arms sticking out into the surround network, which hinders the bead diffusion significantly. Thus, the bead reports larger moduli than without sticking. Equivalently, the effective size of the probe bead is increased due to the filament adsorption. This effect is more profound for smaller particles, since the extent of hindrance due to attached filaments is more significant for smaller beads than for larger beads, given the same filament length. Therefore, smaller beads detect stiffer network as compared with larger beads. Another possible reason for sticky beads sensing higher elasticity is that they tend to draw more filaments towards themselves and hence increase the density of filaments in their proximity, as observed in confocal imaging for carboxylate PS beads (Fig. 3.1). This effect is more striking for smaller beads, when the surface adsorption increases its surrounding filament density more prominently. In both cases, the smaller sticky probes sense a stiffer network than the larger probes. 72 The hopping or percolation effect of slippery beads leads to the positive slope of the measured moduli vs. probe diameter, i.e. smaller beads detect softer networks. When the bead is much smaller than the mesh size, it easily percolates through the network without sensing much hindrance of the networks. When the bead diameter is comparable to or slightly larger than the mesh size of the network, it may be confined in a cage of entangled filaments, yet, it has some probability to hop out of one cage into another. The smaller the bead is, the more chances it has to hop [73], namely, smaller beads experience less network confinement. Similar bead diffusion behavior was first reported by Gardel et al. [66]. They used carboxylate PS beads incubated with G-actin solution before actin polymerization, which we expect to behave similarly to BSA coated beads. In our experiments, however, percolation phenomenon is not observed for BSA beads, suggesting that perhaps a small number of open adsorption sites on BSA coated beads make them slightly sticky and adhere to certain filaments. This remarkable difference implies that the coating of PS beads by monomeric actin prior to polymerization might have blocked the adsorption of actin filaments more thoroughly than that of BSA used in our experiments. The necessity of mixing beads with G-actin before its polymerization in order to detect the hopping effect appears to be an important point not noted in the earlier work [66]. One important feature of probe size dependence is that the measured shear moduli among sticky and slippery beads tended to converge with increasing bead size and actin concentration. The reason for this trend may differ for two categories of probe 73 surface. For the case of sticky beads, physisorption of protein filaments on the bead surface is expected to affect larger beads less significantly than smaller ones. When the network protein concentration increases, the physisorption of the nearby protein filaments might saturate to a level of local concentration less strikingly different from that in the case of a more dilute actin network. For the case of slippery PEG beads, however, a single parameter, namely, the ratio of bead diameter to the mesh size of the network may suffice to describe the effects of both probe size and actin concentration, as suggested by a previous study [73]. Increasing this ratio to much larger than one would effectively suppress the so-called “hopping effect” responsible for the lower measured shear moduli than that of the actual network. In conclusion, the surface effect on microrheology diminishes for both surface categories when beads are much larger than the mesh size and fully encompassed by the network. Depletion of actin filaments near the probe bead surface also affects the applica- tion of particle tracking microrheology. This effect is caused by steric exclusion and is generally applicable to colloidal particles surrounded by macromolecules [114, 115]. The depletion effect creates a layer of lower macromolecular density in the adjacent region of particles. Hence, when the particles probe a surrounding of lower network density, they might detect a softer material compared with the bulk. The thickness of the depletion layer near a flat surface has recently been analyzed for F-actin [75], but prior to this work there has been no quantitative analysis for actin near micron-sized beads. We confirmed the existence of depletion layer around adsorption resistant 74 particles with confocal imaging. The depletion layer thickness is sensitive to both the bead diameter and the length of the surrounding filaments, which is highly polydis- perse in the case of F-actin [116]. Thus, the effect of depletion on microrheology may depend on the probe particle size, but currently there has been no treatment of such an effect available. Due to the limited resolution of the confocal images, we can only qualitatively an- alyze the trend caused by depletion effect on the probe size dependence of microrheol- ogy measurements. For extremely small beads when their diameters are much smaller than the mesh size, there is almost no depletion layer, as illustrated by the small blue bead in Fig. 3.9. When the bead is larger than the mesh size, the depletion effect becomes evident by preventing the filaments from penetrating the surface, though filaments have chances to pass around the bead (e.g. the big bead in Fig. 3.9). When the bead is much larger than the mesh size, the depletion layer thickness approaches that of a flat plane. Thus, the depletion effect is stronger for larger beads, and the depletion layer thickness increases with bead diameter to approach a saturated value. Since there are fewer actin filaments in the depletion layer, larger beads with a thicker depletion layer ought to sense a softer network than the smaller beads. The combined effect of hopping and depletion may be that both small and large beads probe softer networks than some intermediate sized beads. We believe that the depletion effect is secondary to surface stickiness, which causes a more direct coupling between the probe beads and the network. 75 One important insight from this study is the possibility that the size dependence of microrheology might be eliminated by adjusting surface adsorption to the extent that it is totally compensated by the hopping effect, since stickiness and hopping produce opposite trends. In practice, this condition might be achieved by a careful control of the amount of PEG grafted on the bead surface. It has been suggested that shorter PEG chains covering the bead with a high surface density would be more desirable than longer ones for the coating [67], in order to avoid patchy grafts that might complicate the probe. The grafting PEG might also be partially functionalized to preserve weak enough adsorption to compensate for the hopping effect, as has been suggested by Valentine [67]. Slight surface adsorption may also compensate for the depletion effect by pulling in a certain number of filaments towards the probe surface. The approach along this line of thinking is straightforward: one can simply examine the bead size dependence of the scaled MSDs and adjust the relevant parameter accordingly until the size dependence is abolished, i.e. the slope of the scaled MSD or measured shear moduli vs. probe size approaches zero. Our study also provides a cautionary but positive implication on applying particle tracking microrheology within live cells and tissues. On one hand, it is clearly possible to discern the size of the probe particle and all the macromolecular structures the probe might be attached to. On the other hand, since the cellular environment is crowded with proteins, polymers and other soluble molecules, including those that account for the viscoelasticity to be probed, values acquired by measuring the MSD of 76 single probe particles is expected to be less sensitive to the bead surface physisorption than the measurements performed in low protein concentration networks, such as the one illustrated in this study. The mechanical properties probed by particle tracking may be highly variable among different cell types and within different regions of the cell simply because the cellular environment is mechanically diverse and extremely inhomogeneous. For this very reason, particle tracking microrheology remains an effective and reliable technique for detecting local mechanics of living cells and other biological matrix, as well as more traditional soft materials. 3.5 Summary We systematically studied the viscoelastic properties of non-crosslinked actin solutions by tracking the thermal motion of different sized beads with four different surface chemistries, which were characterized by confocal imaging to show descending order of surface stickiness. These four types of beads mainly fall into two categories, slippery beads and sticky beads. They cause opposite trends of probe size dependence of microrheological measurements. For the slippery PEG beads, smaller beads detect softer networks due to the hopping or percolation effect, resulting in positive slopes of the measured moduli vs. probe diameter. By contrast, the other three types of beads are sticky to various extents. Smaller sticky beads detect stiffer networks because they bind to the surrounding filaments, leading to negative slopes. Depletion of actin filaments near the probe bead surface may also affect microrheology in a probe size 77 dependent manner. Our work provides a simple guideline for choosing the optimal probe surface modification: one might eliminate the size dependence of microrheology by adjusting surface adsorption to the extent that it is totally compensated by the hopping effect or depletion effect. Probe size dependence tends to converge for both sticky and slippery beads for larger bead sizes or higher actin concentrations. Thus, our study also provides a cautionary but positive implication on applying particle tracking microrheology within live cells and tissues, since the cellular environment typically has high protein concentrations. Chapter 4 Counterion Dependent Microrheological Properties of F-actin Solution across Isotropic-Nematic Phase Transition 4.1 Introduction In this Chapter, we use particle tracking microrheology to measure the storage (G0 ) and loss (G00 ) shear moduli in the direction parallel and perpendicular to the alignment direction of concentrated actin networks, and characterize anisotropic viscoelasticity of nematic F-actin solution. To our knowledge, this is the first time that particle 78 79 tracking microrheology is applied to the nematic F-actin system. There have been many rheological studies on conventional nematic polymer liq- uid crystals [76, 77, 78, 79]. 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 [80, 81, 82]. Martinoty et al. [82] measured G0 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 G0⊥ larger than G0k . The lowering of G0k in the nematic phase is attributed to the coupling between the shear and the director. Actually, the com- plex shear modulus G∗⊥ and G∗|| for the nematic elastomers can be related to certain components of elastic tensor and viscosity tensor [82, 117]. Though the nematic elas- tomer with covalent crosslinks is different from the nematic F-actin network without crosslinks, these two systems do have similarly anisotropic viscoelasticity. Actin filaments are negatively charged polyelectrolytes. When the filaments are aligned in parallel and close to each other, multivalent counterions, such as M g 2+ , condensed around them can produce an effective attraction between each pair of fil- aments [42, 41, 118]. Such an attractive interaction can induce bundle formation 80 when the counterion concentration reaches a threshold value [48, 40]. This attrac- tive interaction may still exist below the threshold concentration, especially in the nematic phase, though it is not strong enough to suppress the thermal fluctuations and keep the filaments bundled. Consistent with this physical conjecture, we recently reported an anomalous slowdown of F-actin diffusion across the isotropic-nematic (I-N) phase transition caused by the temporary associations between neighboring fil- aments facilitated by divalent counterions [3] (detailed in Chapter 5). Rheological studies on crosslinked actin networks reveal that a very low density of crosslinkers can increase elastic moduli by orders of magnitude [119, 107]. Since these counte- rion induced temporary associations may act like weak crosslinkers, we may ask the following questions. Is the viscoelasticity of F-actin solution affected by the divalent counterions? If so, is it affected differently in the isotropic and nematic phases? We do find a marked influence of divalent counterions on viscoelastic properties in both phases. These findings provide us important insight in how the viscoelastic prop- erties of biopolymeric materials are affected by the polymer alignment, as well as counterion-mediated interactions between the aligned biopolymers. 81 4.2 Materials and Methods 4.2.1 Sample preparation We used carboxylate polystyrene beads with diameters of 1, 2 and 3.6 µm for the study described in this chapter (The exact diameters are specified as 0.984, 2.02 and 3.56 µm by the vendor: Polysciences Inc., Warrington, PA). Bead suspensions were diluted with the same ionic 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 Inc., Mt. Lakes, NJ) with cross-sectional dimensions of 0.1mm×1 mm. Each capillary was immediately sealed with an inert glue to eliminate flow and evaporation from the system. 4.2.2 VPT and LDPT microrheology The expemental setups and the data analysis methods to obtain storage and loss shear moduli from temporal MSD by the VPT and LDPT methods are discussed in more detail in Section 2.3.3.1 and Section 2.3.3.2, respectively. For the VPT method, we tracked the motion of beads embedded in F-actin solution filled in the capillary tubes by using a Nikon E800 upright microscope equipped with a Cool-Snap HQ CCD camera (Photometrics, Tucson, AZ). Time-lapse videos of beads were taken in the middle of the capillary cavity using a 100x objective. The positions 82 of centroids of beads were obtained by using the thresholding tracking algorithm in Metamorph 6.0 (Universal Imaging Corp., Downingtown, PA ). For the LDPT method, we used a Nikon Eclipse TE 2000-U inverted microscope. Laser trap was produced by focusing a laser beam generated by a diode pumped Nd:YAG laser source at 1064 nm infrared wavelength (CrystaLaser LC, Reno, NV). The bead position detection was obtained by a quadrant photodiode detector (custom made by Mr. Winfield Hill, Roland Institute, Cambridge, MA). Voltage outputs for two perpendicular directions were acquired by a BNC 2090 board and processed by LabVIEW (National Instruments, Austin, TX). 4.3 Results 4.3.1 Effects of probe beads on nematic ordering Inclusion of spherical particles in nematic liquid crystals could induce defects [120, 121, 122]. In order to assess for possible distortions and defects of nematic actin network due to the inclusion of spherical particles, we measured the birefringence and the local filament alignment around probe particles using a Polscope retardance imaging system described in Section 2.2.3. The birefringence of nematic actin solution with beads embedded are shown in Fig. 4.1. The sample is measured between glass slide and cover slip sealed with vacuum grease. The average thickness is about 12 ± 2µm, and the average retardance is about 0.99 nm. Fig. 4.1a shows a larger field of 83 a) b) Figure 4.1: Birefringence images of nematic actin solution with beads embedded. The sample is measured between a glass slide and a cover slip. The average sample thickness is 12 ± 2 µm, and the average retardance is about 0.99 nm. (a) A region with beads of three diameters 1 µm, 2 µm and 3.6 µm is shown. The red pins indicate the local orientations of the filaments. The length of each red pin is 0.5 µm, serving as a convenient scale bar. The distortion within the projected areas of beads is due to the birefringence of beads themselves. (b) A zoomed-in image with one 3.6 µm bead at the center is shown. The red pins indicate the local orientations of the filaments. 84 view with test beads of 1 µm, 2 µm and 3.6 µm in diameter. The general alignment is along the horizontal direction, and is not affected significantly by the beads. Fig. 4.1b shows a zoomed-in image around the 3.6 µm bead, which is expected to produce the largest distortion among the three sizes. The distortion of the network is localized over a thin layer around the bead and decays within a micron or two from the bead surface. Also, no defect is observed across the whole sample. Thus, the nematic F-actin network is not severely distorted by micron-sized beads. This conclusion validates our approach to use beads as a probe to study the microrheological properties of nematic F-actin networks. 4.3.2 Anisotropic bead diffusion in the nematic phase The positions of beads at different times were extracted from a time-lapse video recorded by phase contrast microscopy. Fig. 4.2a shows a collection of positions of ten 1 µm beads in a 1 mg/ml isotropic F-actin solution, with 2000 frames per bead taken at the frame rate of 10 frames/sec. The distribution of the bead positions are roughly rotationally symmetric. Fig. 4.2b shows the position distributions of beads in the parallel and perpendicular directions. The distributions in both directions are Gaussian and the widths of the distributions in the two directions are about the same, confirming isotropic diffusion of beads in the isotropic solution. The slightly larger standard deviation in the parallel direction than perpendicular direction is likely due to residual alignment along the capillary axis when the polymerizing F-actin is filled 85 0.4 a) 0.2 r⊥ ( µm) 0 −0.2 −0.4 −0.4 −0.2 0 0.2 0.4 r|| ( µm) 2500 || b) ⊥ 2000 1500 Counts 1000 500 0 −0.4 −0.2 0 0.2 0.4 r|| or r ⊥ ( µm) Figure 4.2: (a) The overlap of positions of 1 µm beads over time in a 1 mg/ml isotropic F-actin solution. Each dot stands for the centroid of a bead in a particular frame, and the graph is a collection of ten beads with 2000 frames per bead taken at the rate of 10 frames/sec. (b) The histograms showing the distributions of bead positions in the parallel (dark bar) and perpendicular (gray bar) directions. The solid lines are Gaussian fits for the distributions. The standard deviations from the fits for the data in parallel and perpendicular directions are 0.14 µm and 0.11 µm, respectively. 86 0.2 a) 0.1 r⊥ ( µm) 0 −0.1 −0.2 −0.4 −0.2 0 0.2 0.4 r|| ( µm) 6000 || 5000 b) ⊥ 4000 Counts 3000 2000 1000 0 −0.4 −0.2 0 0.2 0.4 r|| or r ⊥ ( µm) Figure 4.3: (a) The overlap of positions of 1 µm beads over time in a 4 mg/ml nematic F-actin solution. It is an overlap of trajectories of ten beads with 2000 frames per bead taken at the rate of 10 frames/sec. (b) The histograms showing the distributions of beads in the parallel (dark bar) and perpendicular (gray bar) directions. The solid lines are Gaussian fits for the distributions. The distribution in the perpendicular direction is narrower than that in the parallel direction. The standard deviation from the fit for the parallel direction is about 0.086 µm, and for the perpendicular direction it is about 0.036 µm. 87 into the long capillary. Fig. 4.3a shows a similar collection of positions of ten beads in a 4 mg/ml nematic F-actin solution, and Fig. 4.3b gives the position distributions of beads in the parallel and perpendicular directions. In the nematic solution, the distributions of bead positions in the two directions are also both Gaussian, but the beads diffuse preferentially along the parallel direction (the nematic director). The standard deviation of the distribution in the parallel direction is about 0.086 µm, and in the perpendicular direction it is about 0.036 µm. Hence, the width of the distribution in the parallel direction is more than twice that in the perpendicular direction. From the distributions of bead positions, we see that the bead diffusion in the isotropic solutions is symmetric in all directions, but the bead diffusion in the nematic solution is anisotropic, showing more motion along the direction of alignment. 4.3.3 Mean square displacement and the bead size effect The mean square displacements (MSDs) of beads for different time intervals have been calculated from the coordinates obtained from the VPT method. Fig. 4.4 shows typical MSDs of 1 µm beads in log-log plots for a 1 mg/ml isotropic solution and a 4 mg/ml nematic solution. For the 1 mg/ml isotropic solution, the MSDs in the parallel (¥) and perpendicular directions (¤) are about the same, showing isotropic diffusion. For the 4 mg/ml nematic solution, the bead diffusion is anisotropic. The MSD is larger in the parallel direction (N) than in the perpendicular direction (M), showing preferable diffusion along the nematic director. 88 −2 10 MSD ( µm2 ) −3 10 isotropic || isotropic ⊥ nematic || nematic ⊥ −4 10 −1 0 1 10 10 10 t (sec) Figure 4.4: Mean square displacement (MSD) of 1 µm beads as a function of time interval for 1 mg/ml isotropic solutions and 4 mg/ml nematic solutions. MSDs in the isotropic phase are shown in squares, and MSDs in the nematic phase in triangles. MSDs parallel to the tube long axis are shown in solid symbols, and those perpen- diculer to the tube long axis are shown in open symbols. [Mg2+ ] in either sample is 2 mM. 89 0 10 Modulus (Pa) −1 10 G ’ || G ’⊥ G ’’|| G ’’⊥ −2 10 −1 0 1 10 10 10 ω (rad/sec) Figure 4.5: Frequency dependence of G0 (solid symbols) and G00 (open symbols) measured by the Video Particle Tracking (VPT) method for 1 mg/ml isotropic F- actin solution in the parallel (squares) and perpendicular direction (triangles). The spectrum is obtained by averaging over measurements of ten beads. 4.3.4 Frequency spectrum of shear moduli of nematic F-actin solutions Frequency dependence of shear moduli can be obtained from the MSD via the GSE relation by implementing the power expansion method (Section 2.3.3.1, Eq. 2.2 to Eq. 2.5). In Fig. 4.5, G0 and G00 of parallel and perpendicular directions are plotted against frequency for the 1 mg/ml isotropic F-actin solution containing 2 mM Mg2+ . For the isotropic phase, G0|| and G0⊥ differ very little, and the same holds for G00|| and G00⊥ , showing isotropic viscoelasticity. The small difference between the moduli of two directions could be due to the weak metastable filament alignment in the 1 mg/ml sample caused by the shear flow as the sample was filled into the capillary. The 90 G0 s have a weak dependence on frequency, showing the characteristic of a plateau modulus. The values of G0 obtained are very close to those measured by Gardel et al., between 0.2 and 0.3 Pa for 1 mg/ml F-actin [66]. The G00 s increase with frequency approximately in a scaling relation of G00 ∼ ω 3/4 , consistent with the microrheology measurements by Gittes et al. [64], Xu et al. [60], and Mason et al. [62], and also with the recent optical tweezers passive microrheology results by Brau et al. [111]. The VPT microrheology measurements by Gardel et al. [66], however, gave a scaling relation of G00 ∼ ω 1/2 for a 1 mg/ml F-actin solution using 0.84 µm diameter beads. The extra bump and dip in the low frequency range of G00 spectra are likely artificial, due to smoothing of the noisy data. For the 4 mg/ml nematic F-actin solution containing 2 mM Mg2+ , frequency spectra of G0 and G00 of parallel and perpendicular directions are shown in Fig. 4.6. For the low frequency data on the left, the bead tracking was done with the VPT method and the MSD data were processed with the power expansion method. For the high frequency data on the right, the bead positions were tracked with the LDPT method and the MSD data were converted to the shear moduli via the GSR by directly implementing the fast Fourier transform of the MSDs. For the LDPT method, the trap stiffness is subtracted from G0 s; data of frequencies about 1/2 decade at the lower and higher ends with large truncation errors have been deleted. The moduli measured by the LDPT method are shown in gray symbols in the overlap region. For this particular sample, G0|| , G0⊥ and G00⊥ measured by the two methods agree, evident 91 2 a) 10 Modulus (Pa) 1 10 0 10 LDPT G’ || VPT G’ ⊥ −1 10 2 b) ω3/4 10 Modulus (Pa) 1 10 0 10 LDPT G’’ || VPT G’’⊥ −1 10 −1 0 1 2 3 10 10 10 10 10 ω (rad/sec) Figure 4.6: Frequency dependence of moduli for a 4 mg/ml nematic F-actin solution. (a) shows the spectra of G0|| and G0⊥ in solid symbols; (b) shows spectra of G00|| and G00⊥ in open symbols. For the spectra on the left, the bead position tracking was performed with the VPT method and the MSD was converted to moduli with GSE relation by implementing the power expansion method; for the spectrum on the right, the bead positions were tracked by the LDPT method with the position data converted to moduli via GSR by implementing fast Fourier transform of MSD. In the overlap region between the two dash dot lines, the LDPT results are shown in gray symbols. The two methods give consistent results for G0|| , G0⊥ , and G00⊥ ; G00|| measured by the VPT method, however, is about twice of that measured by LDPT. Both the VPT and LDPT 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 . 92 by the consistent values in the overlap region; G00|| measured by VPT is about twice that measured by LDPT in the overlap region. G0|| and G0⊥ both show the behavior of plateau modulus. G00|| and G00⊥ follow a scaling behavior of ω 3/4 , the same as in the isotropic phase. G0⊥ and G00⊥ are larger than G0|| and G00|| , respectively, across the entire frequency range measured by the two methods. G0⊥ is about three times of G0|| . G00⊥ is either about twice of G00|| as measured by the VPT method, or 4 times of G00|| as measured by the LDPT method. In summary, the viscoelastic properties are isotropic for isotropic F-actin solutions, whereas the nematic F-actin solution is more viscoelastic in a direction perpendicular to the nematic director. The VPT and the LDPT methods give consistent results for the frequency spectra of shear moduli with the exception of G00|| . The cause of such discrepancy is unknown and merits further investigation. 4.3.5 Shear modulus versus actin concentration across the isotropic- nematic phase transition In order to investigate the effect of I-N transition on the rheological properties of F-actin, we have measured G0|| and G0⊥ as a function of actin concentration from 1 mg/ml to 6 mg/ml (Fig. 4.7). The values are taken at 0.3 rad./sec from the fre- quency spectrum for a convenient comparison. The average length of actin filaments is measured to be about 7 µm. The onset concentration for the nematic phase tran- sition is around 2 mg/ml. G0|| and G0⊥ start at about the same value for the 1 mg/ml 93 || ⊥ Figure 4.7: G0|| (solid squares) and G0⊥ (gray triangles) at 0.3 rad./sec are plotted against actin concentration across I-N phase transition of F-actin solution, measured by the VPT method using 1 µm beads. The onset concentration of I-N transition is about 2 mg/ml. The solid line is a power law fit to G0|| , yielding a scaling exponent of 0.54 ± 0.13mesh; the dash line is a power law fit to G0⊥ , yielding a scaling exponent of 1.38 ± 0.15. 94 isotropic solution. As the actin concentration increases and the system enters the nematic phase, G0|| and G0⊥ grow apart, with G0⊥ larger and increasing more steeply than G0|| . The ratio of G0⊥ to G0|| grows from 1 to 4, as the actin concentration in- creases from 1 mg/ml to 6 mg/ml. We find that G0⊥ scales with actin concentration as G0⊥ ∼ c1.38±0.15 , whereas G0|| has a weaker dependence on actin concentration and scales as G0|| ∼ c0.54±0.13 . The scaling behavior of G0⊥ is similar to the bulk rheology results in the isotropic phase G0 ∼ c1.4 [123]. The scaling of G0|| with actin concen- tration in the nematic phase yields a smaller power exponent than that of G0⊥ . This feature is further discussed later. 4.3.6 Dependence of shear moduli on the Mg2+ concentration in isotropic and nematic phases It is expected that [Mg2+ ] may alter the rheological properties of F-actin solution more in the nematic phase than in the isotropic phase. One evidence is that we found F-actin diffusion slowed down by [Mg2+ ] more in the nematic phase than in the isotropic phase [3] (discussed in Chapter 5). We measured for comparison G0|| , G0⊥ , G00|| and G00⊥ as a function of [Mg2+ ] for 1 mg/ml isotropic and 4 mg/ml nematic F-actin solutions (Fig. 4.8). In Fig. 4.8a for the isotropic solutions, G0|| (¨) and G0⊥ (H) are very close for all [Mg2+ ], and the same holds true for G00|| (♦) and G00⊥ (O), showing isotropic viscoelastic properties. In either direction, the G0 s increase with [Mg2+ ] up to 6 mM and then plateaus, and the G00 s are less sensitive to [Mg2+ ]. In Fig. 4.8b 95 a) || ⊥ || ⊥ b) || ⊥ || ⊥ Figure 4.8: G0 and G00 in the parallel and perpendicular directions as a function of [M g 2+ ] measured by the VPT method at 0.3 rad./sec using 1 µm beads. (a) For the 1 mg/ml isotropic solution, G0|| is in ¨, G0⊥ in H, G00|| in ♦, and G00⊥ in O; (b) For the 4 mg/ml nematic solution, G0|| is in ¥, G0⊥ in N, G00|| in ¤, and G00⊥ in M. 96 for the nematic solutions, G0⊥ (N) is larger than G0|| (¥) for all [Mg2+ ], and they both increase with [Mg2+ ] in a larger range up to 16 mM, above which F-actin forms large bundles and the sample becomes highly inhomogeneous. The G00 s are less sensitive to [Mg2+ ], especially G00⊥ . G00⊥ (M) varies little with [Mg2+ ] until the threshold of bundle formation is approached, starting larger than G00|| (¤) but becoming smaller than G00|| at [Mg2+ ] of a few mg/ml. On the whole, the moduli of F-actin solutions are more affected by [Mg2+ ] in the nematic phase than in the isotropic phase. 4.4 Discussion 4.4.1 Anisotropy of viscoelastic properties of nematic F-actin networks Based on the measured MSD, distinct rheological properties are shown between isotropic and nematic actin solutions. For isotropic F-actin solutions, the storage modulus (G0 ) and loss modulus (G00 ) are about the same in the directions paral- lel and perpendicular to the tube long axis across the whole frequency range (Fig. 4.5). The nematic F-actin solutions, however, display anisotropic viscoelasticity, with higher moduli in the direction perpendicular to the alignment of filaments. As shown in Fig. 4.6, G0⊥ and G00⊥ are larger than G0|| and G00|| , respectively, over the entire frequency range. The anisotropic bead diffusion and viscoelasticity of nematic F-actin solutions can 97 be interpreted based on the tube model [25, 124], which describes the motion of an individual filament in an entangled network as diffusing in a virtual tube constructed by its neighboring filaments. Consequently, the transverse diffusion of the filament is significantly suppressed, whereas its longitudinal diffusion is mostly unhindered. When the network becomes nematic, we propose that the motion of a single filament can still be modeled as moving in a tube, although the tube is dilated to a certain extent compared with the isotropic phase due to the alignment of filaments [25]. Since the transverse diffusion of the filaments is suppressed, the filaments tangential to the bead (e.g., filaments #1 and #2 in Fig. 4.9b) experience more restriction in transverse motion. Consequently, they can hardly vacate room for the bead to diffuse. The filaments pushing the bead with their ends (e.g., filaments #3 and #4 in Fig. 4.9b), however, can more easily diffuse away longitudinally, hence leaving more room for the beads to diffuse. Consequently, the beads diffuse more along the alignment of actin filaments. The elasticity of non-crosslinked actin network in the linear regime is mostly at- tributed to bending of filaments (enthalpic elasticity) and longitudinal extension or compression of filaments (entropic elasticity) [125]. In nematic F-actin solutions, the bead is embedded in a local arrangement of filaments as illustrated in Fig. 4.9b). The filaments contacting the bead tangentially (e.g., #1 and #2 ) push the bead in the direction normal to the filament axis via bending of their own contours. Since 98 they are generally aligned while entangled with one another, they can work cooper- atively and the elasticity sensed by the tracer bead in the perpendicular direction is therefore strong. The filaments making contact with their ends to the bead (e.g., #3 and #4 ) push or pull the bead by extending or compressing their contours. Because these filaments are not anchored, they can readily slide against each other, and hence provide less elastic support. Thus, the elasticity in the parallel direction is weaker than that in the perpendicular direction. 4.4.2 Dependence of shear moduli on actin concentration for nematic F-actin solutions Dependence of shear moduli on actin concentration for isotropic solutions has been measured by several research groups [64, 59, 62, 63, 60]. As far as we know, the work reported here is the first attempt to measure the shear moduli of F-actin solutions in the nematic phase using the VPT microrheology. In the isotropic phase (with actin concentration below 2 mg/ml), we found that G0⊥ is equal to G0|| , showing isotropic viscoelasticity (Fig. 4.7). When the system enters the nematic phase with increased actin concentration, G0⊥ becomes larger and increases more steeply than G0|| . G0⊥ scales with actin concentration as G0⊥ ∼ c1.38±0.15 , whereas G0|| depends more weakly on actin concentration and scales as G0|| ∼ c0.54±0.13 . The ratio of G0⊥ to G0|| grows from 1 at 1 mg/ml actin concentration to about 4 at 6 mg/ml. This is in the same order of magnitude as reported by [82] for nematic elastomers, which is a different type of 99 material. The G0⊥ /G0|| ratio of nematic elastomers is about 1.6 at the transition point. G0⊥ in the nematic phase scales very similarly to the plateau modulus G0 measured in the isotropic phase. The bulk rheology measurement by Hinner et al. [123] yields G0 as a function of actin concentration for isotropic F-actin solutions, following G0 ∼ c1.4 . Slightly different in exponent, the diffusing wave spectroscopy microrheology of Xu et al. [60] gives G0 ∼ c1.2 and the VPT microrheology of Gardel et al. [66] gives G0 ∼ c1.8 for isotropic F-actin solutions. The theoretical form of plateau modulus of isotropic entangled semiflexible polymer networks has been obtained by Isambert & Maggs [126] and Morse [124] as G0 ∼ ρkB T /le ∼ c7/5 /lp1/5 , (4.1) where ρ is the number density of the polymers, c is the mass concentration in mg/ml, 1/5 lp is the persistent length, and le is the entanglement length. le ∼ ξ 4/5 /lp , with ξ being the mesh size and ξ = 0.3 c−1/2 [113]. If G0⊥ in the nematic phase follows the same scaling behavior as G0 in the isotropic phase, we would expect that the rationale deriving Eq. 4.1 still holds, which means that entanglement still exists in the nematic phase and le scales with concentration with the same exponent. As the filaments align in the nematic phase, the mesh can be understood as elongated along the nematic director and the segments constructing the mesh become longer. Since the mesh in the nematic phase becomes asymmetric along and perpendicular to the alignment, it may require two parameters to describe the mesh size, i.e. ξ|| and ξ⊥ (dimensions along and perpendicular to the alignment) with ξ|| > ξ⊥ . The perpendicular elasticity 100 sensed by the probe beads is still caused by an entangled filament mesh. Thus, G0⊥ remains inversely proportional to le (Eq. 4.1), which is then related to ξ|| and ξ⊥ . Although the ratio of ξ|| to ξ⊥ depends on the nematic order parameter, both ξ|| and ξ⊥ vary with actin concentration similarly to ξ in the isotropic phase. Hence, we expect a similar scaling relation of G0⊥ with actin concentration. The physical picture put forth here reinforces the concept of entanglement in the nematic phase. This concept has recently been introduced based on the experimental finding that the order parameter of nematic F-actin solutions is significantly smaller than 1 [34]. G0|| in the nematic phase scales approximately as G0|| ∼ c0.5 . This has been mea- sured for the first time, as far as we know. If the G0|| is mostly due to the pushing of the filaments by their ends (filaments #3 and #4 in Fig. 4.9b), one would expect G0|| to be proportional to c and the scaling exponent to be 1. One explanation to this 0.5 exponent is that due to the increase of actin concentration the filaments become more and more aligned, enhancing the sliding motion of filaments with respect to each other. This effect reduces the increase of elastic support due to the increase of the number of pushing filaments. Therefore, G0|| is smaller than being proportional to c. An alternative explanation is that the cross-section of the virtual cylinder (between the two dashed lines in Fig. 4.9b) accommodating the filaments contributing to the elastic support in the parallel direction shrinks due to the increased osmotic pressure on the cylinder with the increased actin concentration. The elastic support in the 101 parallel direction should be proportional to the cross-section area times actin con- centration. With a decreasing cross-sectional area, consequently, we see the scaling behavior of G0|| on actin concentration with an exponent smaller than 1. 4.4.3 Viscoelasticity of F-actin solutions altered by Mg2+ con- centration In the isotropic phase, the storage shear moduli increase with [Mg2+ ] up to 6 mM and then saturate (Fig. 4.8a). There are also 50 mM K + and 0.2 mM Ca2+ in the actin solution, ensuring predominant actin polymerization under all Mg2+ concentrations. The dependence of [Mg2+ ] can be interpreted by the temporary association between segments of two neighboring filaments especially when they are parallel. The effect has been detected as the slowdown of F-actin diffusion in the nematic phase with in- creasing [Mg2+ ] [3]. These temporary associations are likely caused by the counterion mediated attraction between neighboring filaments at their junctions facilitated by the condensed counterions. These temporary associations act like weak crosslinkers, which increase the elasticity of the network (Fig. 4.9a). In the nematic phase, both G0|| and G0⊥ increase with the increase in [Mg2+ ] over a wider range (Fig. 4.8b). The temporary associations facilitated by condensed Mg2+ ions are more favorable in the nematic phase than in the isotropic phase, since the filaments are mostly aligned in the nematic phase. These associations cause parallel filaments like those illustrated on the top and bottom of the bead in Fig. 4.9b 102 a) + + + ++ + + + + X b) + ++ ++ 2 + ++ + + + + ++ 1 ++ + + + + ++ ++ ++ ++ + ++ 3 4 ++ ++ ++ ++ ++ + + + + + ++ + Figure 4.9: Illustrations of beads embedded in isotropic (a) and nematic (b) actin networks. The curved lines are actin filaments. “+”s, each enclosed by a circle, represent M g 2+ ions. (a) A micron sized bead embedded in an isotropic actin network. The mesh size in this condition is ξ = 0.3 c−0.5 = 0.3 µm, where c = 1 mg/ml. The M g 2+ ions only occasionally associate two filaments over their parallel parts or at their junctions. (b) A micron sized bead embedded in a nematic actin network. The average spacing of two neighboring filaments estimated using a hexagonal lattice model is d = 0.17 c−0.5 = 0.085 µm, where c = 4 mg/ml. The M g 2+ ions more frequently associate neighboring filaments in the nematic phase, since they are mostly parallel to each other. 103 (e.g., filaments #1 and #2) to bend more collectively, thus increasing G0⊥ . However, these associations do not contribute strongly to the energy dissipation of transverse motion of the bead. Hence, G00⊥ is not significantly affected by [Mg2+ ]. In the parallel direction, the associations help immobilize the relative sliding between filaments, causing the filaments on left and right of the bead (e.g., filaments #3 and #4 in Fig. 4.9b) to provide more elastic support, and thereby increasing G0|| . Considering that Mg2+ causes increase in the friction between neighboring parallel filaments [3], it is intriguing to note that G00|| increases with [Mg2+ ] only moderately. We conclude that the beads probe more directly the dissipation of the neighboring network, whereas the single filament motion is affected by the interaction with its neighboring filament more sensitively. 4.5 Summary We apply both VPT and LDPT microrheology methods to probe viscoelastic prop- erties of F-actin across the I-N phase transition region. The two methods give con- sistent results for G0 and G00 . For isotropic F-actin solutions, the viscoelastic moduli are isotropic; in the nematic phase, F-actin solutions are more viscoelastic in the direction perpendicular to the nematic director. At low actin concentrations, G0|| and G0⊥ are equal. As the actin concentration increases above the onset concentration of I-N transition, G0|| and G0⊥ grow apart from each other, with G0⊥ larger than G0|| . G0|| scales with actin concentration weakly as G0|| ∼ c0.54 . G0⊥ follows a scaling relation 104 of G0⊥ ∼ c1.38 , nearly the same as the plateau modulus in the isotropic phase. In the nematic phase, parallel filaments bend cooperatively due to the remaining entangle- ment, contributing to stronger viscoelasticity in response to transverse deformation, whereas the filaments can readily slide against each other along the nematic director, thus leading to reduced viscoelasticity along the direction of alignment. Furthermore, we studied and compared the effect of [Mg2+ ] on G0 and G00 for isotropic and nematic F-actin solutions. For the isotropic phase, G0 increases with [Mg2+ ] in a limited range and then plateaus. For the nematic phase, both G0|| and G0⊥ increase progressively with [Mg2+ ] in the whole range below the F-actin bundling threshold concentration, which is about 16 mM. The increase of G0 s is attributed to the temporary association between neighboring filaments induced by the condensed Mg2+ ions. The increase of G0 is more pronounced in the nematic phase since the Mg2+ mediated attraction is more tangible between parallel filaments. In both isotropic and nematic phases, G00 only weakly depends on [Mg2+ ]. In conclusion, the study of microrheological properties of F-actin solutions using particle tracking methods across the I-N phase transition under various Mg2+ concen- trations unites the pertinent effects of alignment and electrostatic interaction between the protein filaments. The results of this study provide a valuable example that the microrheological approach leads to useful insights on the viscoelastic properties of a wide range of biological and synthetic polymeric materials. Chapter 5 Counterion Induced Abnormal Slowdown of F-actin Diffusion across Isotropic to Nematic Phase Transition 5.1 Introduction In this Chapter, we study the motion of F-actin inside highly entangled isotropic and nematic networks. The tube model and concept of reptation were first introduced by P. G. de Gennes in 1971, when he studied the possible motion of a polymer chain inside a strongly crosslinked polymer gel [127]. The topological constraint that a polymer chain cannot intersect any other chains dictates that the polymer is confined 105 106 in a virtual tube defined by its neighboring filaments. The polymer diffuses by repta- tive motions along the tube with its ends constantly exploring paths to another tube. (See Fig. 5.1) The theoretical models of Doi and Edwards based on the de Gennes model provided the first molecular model of the polymer system for the two extreme cases of completely flexible and rigid rods [25]. The reptation tube model have also proven to be highly successful in describing the motion of semiflexible macromolecular filaments in concentrated solutions, including F-actin [128, 123, 129, 130, 131]. Käs [132, 29] implemented a direct fluorescence imaging of the reptation of actin filaments in a virtual tube, and they quantified the average tube diameter and the longitudinal diffusion coefficient of filaments. David C. Morse [124] recently performed a theo- retical study of the behavior of semiflexible chains in the tightly-entangled solutions, −1/5 yielding the average tube diameter De ≈ 1.6(cL)−3/5 Lp with a binary collision ap- proximation (BCA), or De ≈ 0.84(cL)−1/2 with an effective medium approximation (EMA), where c is the number concentration of polymers per unit volume, L is the polymer contour length, and Lp is the persistence length. Since high extent of entan- glement is preserved in the nematic F-actin network, the reptation tube concept also applies to F-actin in the nematic phase. In this work, we report a surprising slowdown of longitudinal diffusion (tube repta- tion) of F-actin across the region of isotropic to nematic (I-N) phase transition, based on a microscopic study of filament motion. For most rod-like systems, such small molecule liquid crystals [133], liquid crystal polymers [134], and rod-like biopolymer 107 Figure 5.1: Illustration of the reptation tube model for Factin. The filament (black curve) is confined to the reptation tube due to the fact that it cannot penetrate the neighboring filaments (black circles). The reptation tube is depicted by the dashed curves. The reptative motion of the filament ends allows for exploring new reptation tubes as shown by the right end of the filament. solutions [135], the longitudinal diffusion enhances sharply at the I-N transition point. The origin for an increased diffusion in the nematic phase for the rod-like suspensions is usually attributed to the tube dilation effect [25]. To interpret this counter-intuitive slowdown of F-actin in nematic phase, we compared the diffusion of F-actin, micro- tubule and fd virus in F-actin solutions across the transition region, and measured the dependence of viscous drag probed by F-actin on [M g 2+ ] in the isotropic and nematic state. Based on these findings, we propose that the abnormal slowdown is caused by the transient associations between actin filaments mediated by divalent counterions, which is more striking when filaments are aligned in parallel. 108 5.2 Experimental Methods 5.2.1 Sample preparation The actin samples were prepared to have three different average filament lengths of 2.7, 4, 7 µm. The batch used in the experiments have an average filament length of 7 µm under the specified polymerization condition. The average length of F-actin was regulated by adding gelsolin to G-actin before initiating polymerization. For instance, the average filament lengths of 2.7 and 4 µm was calculated as ¯l(µm) = rAG /370 [116], where rAG is the molar ratio of G-actin to gelsolin. Small amount of F-actin solution was taken out to be fluorescently labeled by Rhodamine Phalloidin (Cytoskeleton, Inc., Denver, CO), and actin monomers were about 70% labeled. Then, the labeled F-actin were diluted and added to the original solution in 2% volume fraction. The final ratio of labeled to unlabeled filaments was below 1:1000 to enable single filament tracking. Antiphotobleaching reagents were added to samples to reach final concentrations of 4 µg/ml catalase, 0.1 mg/ml glucose, 20 µg/ml glucose oxidase, and 0.05 vol % mercaptoethanol, allowing for long observation time [29]. The mixed solutions were injected into rectangular capillary tubes (VitroCom Inc., Mt. Lakes, NJ) with cross-sectional dimensions of 0.05mm × 0.5mm. Each capillary was immediately sealed with an inert glue to eliminate flow and evaporation from the system. 109 5.2.2 Single filament tracking method To track the centroid position of a single filament as time evolves, we first acquire a time lapse video (in “.stk” format) of fluorescently labeled filaments diffusing in unlabeled background F-actin solutions using MetaMorph 6.0. The video typically has a duration of 10 seconds and a frame rate of 10 frames/second. A small region covering the diffusion of a single filament is cropped from the original video. This smaller video is then export to a series of “.tif” image files with serial file names. Next step is to use MatLab [136] to analyze each image to find the centroid of filament in each frame, with the detailed procedures as follows (MatLab code in Appendix): 1. The original image is loaded and converted to grey scale from RGB format. (Fig. 5.2a) 2. The grey scale image is converted to a binary image (with only black and white), by thresholding the image using the MatLab function ‘imextendedmax’. If the pixel intensity is larger than pre-selected threshold value, the pixel value is set to one; otherwise, zero. (Fig. 5.2b) 3. The binary image is processed to smooth the edges and remove all the black pixels in the white region, using the Matlab function ‘bwmorph’ with the pa- rameter ‘close’ and ‘open’. (Fig. 5.2c) 4. The center thread of the filament with width of a pixel is obtained using the Matlab function ‘bwmorph’ with the parameter ‘thin’. (Fig. 5.2d) 110 Figure 5.2: (a) A grey scale fluorescent image of an actin filament. (b) A binary image of the filament after thresholding. (c) The binary image after smoothing the edges. (d) Center thread of the filament. The centroid is stamped as a green cross. 5. The centroid of filament is located with the Matlab function “regionprops”, and stamped on the image with a green cross. The centroid is usually not on the filament. After obtaining the coordinates of the centroid of filament for every frame, we perform a coordinate transformation to rotate the coordinates and make the average orienta- tion of the filament aligned horizontally. Thus, the X (horizontal) and Y (vertical) coordinates reflect the longitudinal and transverse motions, respectively. The average orientation can be obtained by linearly fitting the filament contour. The longitudi- nal and lateral coordinates are then used to compute the temporal MSD, diffusion coefficient, and microscopic viscosity of the network. 111 5.2.3 Birefringence measurement using Polscope The order parameter of F-actin solution was quantified by birefringence measure- ments. The specific retardance (in the unit of nm · ml/mg) of F-actin solutions was measured using a LC-Polscope system described in Section 2.2.3 (LC-Polscope, Cam- bridge Research and Instrumentation, Woburn, MA). The polarizing microscope, a Nikon Eclipse 800, was operated as described in the literature [89, 91]. The LC- Polscope software was used for instrument control, image capture, and computation of retardance and orientation maps. The LC-Polscope system is capable of measuring the optical retardance and orientation of the slow axis (alignment of actin filaments) at each pixel position. The specific retardance was obtained by the average retar- dance (in the unit of nm) in the field of view divided by the concentration of actin solution. 5.3 Results and Discussion 5.3.1 Single filament reptation in a virtual tube To gain insight into the motion of F-actin in concentrated solutions, we tracked the motion of a small number of fluorescently labeled actin filaments in unlabeled back- ground F-actin solutions. From a time-lapse fluorescence video, the coordinate in- formation of the centroids of individual filaments can be extracted (with detailed 112 a) b) Figure 5.3: (a) Overlay of the center threads of a single actin filament from a series of time-lapsed images. The white thread shows the initial position of the filament, with the later configurations added as black threads. This filament shows primarily a 1-D diffusion, as if being confined within a virtual tube. (b) A typical plot of the longitudinal diffusion coefficient of individual filaments v.s. filament length (¯). This solution has a concentration of 5.14 mg/ml and an average filament length ¯l = 2.7 µm. The dotted curve is the fit to Eq. 5.1 to obtain ηm . In this case, we find η to be 2.69 cp. The inset shows the longitudinal MSD of two representative filaments. The fit to (∆X)2 = 2Dt gives the diffusion coefficients for 1.9 µm (M) and 6.9 µm (¥) filaments, 0.72 µm2 /sec and 0.30 µm2 /sec, respectively. 113 procedures described in Subsection 5.2.2). The lateral displacement is largely sup- pressed by the surrounding network of actin filaments, while the longitudinal displace- ment is approximately an order of magnitude greater than the lateral displacement (Fig. 5.3a). In other words, the actin filament moves as if it reptated in a virtual tube confinement [127]. In this work, we focus on the longitudinal motion of F-actin. From the reptation of the filaments, we can obtain the longitudinal diffusion co- efficients of different filaments, and the microscopic viscosity of the network. Typical longitudinal mean square displacements (MSD) as a function of time are shown in the inset of Fig. 5.3b, for two filaments with contour lengths of 1.9 and 6.9 µm in a solution with an actin concentration of 5.14 mg/ml and ¯l =2.7 µm. The MSD follows the 1-D diffusion relation. By fitting the data of MSD v.s. time to (∆X)2 = 2Dk t, we obtain the longitudinal diffusion coefficients for these two filaments, which are 0.72 µm2 /sec and 0.30 µm2 /sec, respectively. Performing the same analysis for filaments of different lengths, we obtain the diffusion coefficient as a function of filament length (Fig. 5.3b). The drag coefficient ζ in the dilute limit of a straight rod is ζ = ρπηL/ln( Ld ), where ρ is either 2 for longitudinal or 4 for lateral motions, respectively, η is the solvent viscosity, L is the filament contour length, and d is the filament diameter. The fit of the data to the Stokes-Einstein relation, Dk = kB T /ζ = kB T ln(L/d)/2πηL (5.1) with η as the only fitting parameter yields an microscopic viscosity, e.g., ηm = 2.69 114 a) b) Figure 5.4: (a) Microscopic viscosity ηm as a function of actin concentration across the I-N phase transition region is shown for three ¯l values of 2.7, 4 and 7 µm. All three series of samples show a marked increase of ηm in the transition region. (b) The specific retardance is plotted against actin concentration across the same region. The increase of the order parameter (proportional to the specific retardance) is shown to be concurrent with the rise of ηm in the transition region. cp for Fig. 5.3b. It is worth noting that ηm only applies to longitudinal diffusion of F-actin, and is very different from the bulk viscosity for an F-actin solution. 5.3.2 Abnormal slowdown of filament diffusion across the isotropic to nematic phase transition We find an abnormal slowdown of longitudinal filament diffusion across the I-N tran- sition region, after measuring a range of actin concentrations. Microscopic viscosity ηm is plotted against actin concentration across the transition region for ¯l values of 115 2.7, 4 and 7 µm (Fig. 5.4a). We also determined the transition region by measur- ing the optical birefringence for samples of corresponding ¯l. The specific retardance, which is proportional to the orientational order parameter, is shown in Fig. 5.4b. A close comparison between Fig. 5.4a and 5.4b indicates that a marked increase in ηm occurs in the transition region for all three ¯l values. Samples with a shorter ¯l have a smaller increase. This means that the longitudinal diffusion slows down as the phase transition proceeds, which is abnormal and counter-intuitive. In length of several- contrast to F-actin, for small molecule liquid crystals [133], rod-like polymer systems such as PBLG [134], and fd virus solutions (¯ in Fig. 5.5 ), the longitudinal ηm decreases sharply in the I-N transition region. The increased longitudinal diffusion in the nematic phase for the rod-like system is usually attributed to the tube dilation when the rods are aligned [25], which reduces the drag between the filament and the virtual tube wall defined by the surrounding filaments. 5.3.3 Counterion mediated temporary association accounts for the slowdown To find the mechanism of this abnormal slowdown effect of F-actin, we compared the diffusion behaviors of F-actin, microtubules (MTs) and fd virus in F-actin solutions across the I-N transition region. Fd viruses are semiflexible filaments with a length of 0.88 µm, diameter 6.6 nm, and persistence length 2.2 µm [137, 138]. MTs, another 116 135 Figure 5.5: The comparison of ηm of F-actin (¥), microtubules (F) and fd virus (♦) in F-actin solutions with 50 mM KCl and 2 mM M gCl2 across I-N phase transition. Also, the data of fd diffusion in fd solution, converted from Ref. [135], is presented as a comparison (¯). In addition, ηm of F-actin in F-actin solutions with 50 mM KCl but no M gCl2 is also shown (N). The concentrations are normalized with respect to the phase transition concentration. The normalized concentration of 1 stands for 2 mg/ml for F-actin and 15.5 mg/ml for fd, respectively. 117 major component of the cytoskeletal network, have a diameter of 24 nm and a per- sistence length of several mms [139, 28]. Fd viruses were fluorescently labeled with rhodamine; MTs were labeled with oregon green taxol. The tracking of fd viruses or MTs in F-actin solution was carried out with the same procedure as that of F-actin. Fig. 5.5 shows ηm probed by F-actin, MT and fd virus as a function of actin con- centration spanning the F-actin I-N transition region. These experiments were done using a background F-actin solution with ¯l =7 µm. In Fig. 5.5, ηm of F-actin increases over 300% across the transition region. The increase of either fd virus or MT is about 20∼30%, much smaller than that of F-actin. The different diffusion behaviors of these three probe macromolecules are likely caused by the difference among the interactions of actin-actin, MT-actin, and virus-actin. We hypothesize that the electrostatic interaction between neighboring filaments is a key factor in the slowdown of the probe filaments, and the different charge properties cause the probe MT and fd to be affected much less. To support this idea, we measured ηm of F-actin v.s. actin concentration in solutions without divalent counterions M g 2+ (Fig. 5.5). There is a substantial difference between F-actin solutions with (¥) and without M g 2+ (N). With 2 mM M g 2+ , the increase in ηm of F-actin is 3 times across the transition region. Without M g 2+ (there is always 0.2 mM Ca2+ in solution), the increase is reduced to under 50%, and the dependence of ηm on actin concentration is very similar to that probed by MTs (F). The slowdown effect diminishes without M g 2+ . This means that the abnormal slowdown is facilitated by the presence of 118 divalent counterions. The physical picture we propose is that the counterion mediated attraction be- tween charged filaments (polyelectrolytes) results in the slowdown of filament diffu- sion in the nematic phase. When the filaments are parallel and very close, multivalent counterions, such as M g 2+ , condensed around these filaments can produce an effective attraction between them [42, 41]. Such an attractive interaction can induce formation of bundles when the counterion concentration reaches a threshold value [48]. Below that concentration, the attractive force is not strong enough to suppress the thermal fluctuations and keep the filaments bundled. It can, however, make neighboring fil- aments form a temporarily bound state when they approach the interacting range. Fig. 5.6a illustrates the attraction between parallel actin filaments in the nematic phase. The divalent counterions act like bridges, causing two neighboring filaments to be attracted to each other and partially associate momentarily. In the isotropic phase, the filaments exist in random orientations, so the counterion mediated attrac- tion could only occur at their junction, which may be disrupted by the Brownian motion. Thus, this mechanism explains the concurrency of the slowdown of longitu- dinal diffusion with the orientational order. This picture can qualitatively explain the different diffusion behaviors of F-actin, MT and fd virus in F-actin solutions, based on the different charge densities of the three kinds of filaments. MTs and fd virus have more negative charges per unit length. They are more inclined to be repelled by actin filaments. It is known experimentally, 119 for instance, that fd virus is less sensitive to [M g 2+ ]: F-actin forms bundles at [M g 2+ ] about 25 mM [48], whereas fd virus forms bundles at about 60 mM M g 2+ [140]. In addition, since fd viruses are shorter and more flexible, it may be entropically more unfavorable for them to laterally associate with adjacent filaments [141]. This picture can also explain the result that shorter F-actin shows less slowdown (Fig. 5.4a). Since shorter filaments are more inclined to rotate and less likely to accommodate parallel association, they experience less drag per unit length and weaker slowdown effect. Note in Fig. 5.5 that for MT (F) or fd virus (♦) diffusion in F-actin solution and F- actin diffusion without M g 2+ (N) there is consistently a small slowdown effect instead of an increased diffusion as expected for rodlike systems across the I-N transition region. We think that due to their long length and flexibility, F-actin in the nematic phase actually forms an entangled network. One evidence is the imperfect alignment of F-actin in the nematic phase, with the order parameter significantly smaller than 1 [34]. Perhaps because of this imperfect alignment and entanglement in the nematic phase, the probe filaments continuously experience more drag with the increase of actin concentration even in the transition region. Thus, the entanglement diminishes the expected increase in the longitudinal diffusion of F-actin over the I-N transition region, whereas the steep increase is due to the electrostatic interaction mediated by divalent counterions. 120 a) - ++ - - - ++ ++ ++ - - - - - - - ++ ++ - - - - - - ++ ++ ++ ++ - - - - - - - - - ++ ++ ++ b) Figure 5.6: (a) The illustration of weak attraction between actin filaments induced in the nematic phase by divalent counterions. ‘-’ represents negative charges on F-actin, and ‘++’ represents divalent counterions. The dark filament is confined in a virtual tube defined by its surrounding filaments. The divalent counterions temporarily associate parts of the filaments. (b) ηm of F-actin in the nematic phase (4 mg/ml) is plotted against [M g 2+ ] (¥). The data in the isotropic phase (1 mg/ml) is also shown (4). The fit to our model is given in dashed line. 121 5.3.4 A simple model Based on this picture, we propose a simple model to explain how the slowdown phenomenon is affected by [M g 2+ ] in the nematic phase. The simplifying assumption of the model is that once a filament laterally associates with neighboring filaments, it stops moving for a time interval τ , which is assumed to follow an Arrhenius type behavior, i.e. τ = τ0 exp(E(CM g )/kB T ) , (5.2) where E is the activation energy for the filament to break the temporary binding. Assuming E ∝ CM g , Eq. 5.2 becomes τ = τ0 exp(A · CM g ) , (5.3) where A is a prefactor. The actual diffusion time is obtained by subtracting the time being stationary from the total time. The longitudinal diffusion equation becomes (∆X)2 = 2Dk0 (t − nt · τ ) = 2Dk t , (5.4) where Dk0 is the diffusion coefficient without M g 2+ and n is the collision rate. From Eq. 5.4, we obtain an effective longitudinal diffusion coefficient Dk = Dk0 (1 − nτ ) . (5.5) Then, the effective microscopic viscosity ηm is related to the diffusion coefficients as ηm = (D||0 /Dk )ηm0 = ηm0 /(1 − nτ ) . (5.6) 122 Finally, substituting Eq. 5.3 into Eq. 5.6, we obtain ηm = ηm0 /(1 − nτ0 exp(A · CM g )) . (5.7) To test this model, we further measured ηm of F-actin with different [M g 2+ ] in both isotropic and nematic phases. The dependence of ηm on [M g 2+ ] was measured for 4 mg/ml nematic F-actin solutions and 1 mg/ml isotropic solutions below the bundling concentration (about 20 mM) (Fig. 5.6b). There is a 4 fold increase in ηm in the nematic phase when [M g 2+ ] rises from 0 mM to 17 mM. In contrast, ηm shows little change with the increase of [M g 2+ ] in the isotropic phase. This means that [M g 2+ ] plays an important role in affecting F-actin diffusion in the nematic phase, but not in the isotropic phase. Fitting the data in the nematic phase to Eq. 5.7 gives the values of the fitting parameters ηm0 = 3.78 cp, nτ0 = 0.41 and A = 0.04 mM−1 (dashed line in Fig. 5.6). With the fitted values of nτ0 and A, by setting nτ0 exp(A · CM g ) = 1, we find CM g = 22 mM, which corresponds to the threshold concentration for bundle formation. This value agrees well with the experimental value of 20 mM under the ionic condition of this study, correctly predicting the onset of bundle formation. We can further roughly estimate the collision rate n based on the reptation dy- namics and accordingly yield an estimate of τ0 . First, we assume that the filament contour takes a sinusoidal form with an amplitude of the tube radius R, about 0.02 µm for 4 mg/ml solution [113]. Integrating the bending energy over the filament 123 contour and assigning a bending energy of kB T per wavelength λ of filament, we find λ ≈1.5 µm. Presumably, the filament can be divided into segments with length λ, which laterally diffuse independently. The time for each segment to diffuse over a tube diameter is tD = 4R2 /D⊥ , where D⊥ is the lateral diffusion coefficient, and D⊥ = kB T ln(λ/d)/4πηλ, with η being the viscosity of water (1 cp). Then, the col- lision rate n = (1/tD ) × (¯l/λ), which is about 3500 s−1 for 4 mg/ml F-actin with ¯l =7 µm, so τ0 is on the order of 0.1 ms. Based upon this estimation, we can also qualitatively explain the average length effect, i.e. F-actin with shorter ¯l has a smaller increase in ηm across the transition region. A smaller ¯l corresponds to a smaller n, resulting in a smaller ηm from Eq. 5.7. 5.4 Summary We discovered an abnormal increase in ηm of F-actin across the I-N phase transition. Furthermore, an increased [M g 2+ ] results in a larger slowdown. Entanglement in F- actin over the I-N transition region abolishes the expected increase in the longitudinal diffusion. The abnormal slowdown of F-actin is proposed to be primarily caused by the divalent counterion mediated attraction in the nematic phase. A simple model based on this picture quantitatively explains the effect of [M g 2+ ] on the slowdown phenomenon and qualitatively accounts for the different behaviors among different probe filaments. Chapter 6 Summary of Thesis This thesis presents a microscopic investigation of local mechanical properties of F- actin networks via single particle tracking microrheology and a single filament track- ing technique. Both methods probe the material properties by tracking the thermal motion of the probe objects; the former method uses micron-sized spherical parti- cles as probes to measure the shear moduli, whereas the latter is non-perturbative in the sense that it takes advantage of the filaments themselves as probes to obtain a microscopic viscosity. This work implements these two methods, and examines in detail how microscopic network properties of F-actin are affected by the surface chem- istry of probe particles, liquid crystalline phase transition, and counterion mediated interaction. First, we investigate how mechanical properties of F-actin solution measured by single particle tacking microrheology depend on the probe surface properties and 124 125 probe size. We use four types of beads, including carboxylate PS beads, silica beads, BSA coated PS beads and PEG grafted PS beads, which are characterized by con- focal imaging combined with particle tracking to have descending surface stickiness. Different stickiness leads to different probe size dependence of measured shear mod- uli. For the most slippery PEG beads, the smaller beads sense a softer network, since smaller beads have more chances to percolate through the network or hop from one confinement “cage” to another. The other three types of stickier beads follow an op- posite trend of probe size dependence with the smaller beads sense a stiffer network, due to the physisorption of nearby filaments to the bead surface. We also confirm the existence of a depletion layer around non-sticky or weakly sticky probe surfaces by confocal image analysis. Analysis of the effects of surface adsorption, probe hopping and depletion is necessary in order to precisely interpret microrheology data. Opti- mal probe particle to minimize these artifacts can be achieved by properly coating the particle surface so as to balance the opposite effects produced by adsorption and hopping. Furthermore, we study rheological properties of F-actin across the isotropic-nematic phase transition region by particle tracking microrheology. The viscoelastic properties are isotropic in the isotropic phase, whereas in the nematic phase F-actin solutions are more viscoelastic in the direction perpendicular to the nematic director. G0|| and G0⊥ start off at the same value in the isotropic phase, and then G0⊥ grows larger and faster than G0|| as the system enters nematic phase with the increase of actin concentration. 126 The storage moduli scale with actin concentration as G0⊥ ∼ c1.38 and G0|| ∼ c0.54 , re- spectively. In the nematic phase, parallel filaments have to bend in coordination due to the remaining entanglement, contributing to stronger viscoelasticity in response to transverse deformation, whereas the filaments can readily slide against each other along the nematic director, thus leading to reduced viscoelasticity along the direction of alignment. Also, we compare the effect of [Mg2+ ] on G0 and G00 for isotropic and nematic F-actin solutions. The increase of shear moduli with [Mg2+ ] is more striking in the nematic phase than in the isotropic phase. This phenomenon is attributed to the temporary association between neighboring filaments induced by the condensed Mg2+ ions, which is more prominent in the nematic phase when filaments are aligned in parallel. Finally, we study the diffusion dynamics of F-actin using fluorescence imaging and a single filament tracking technique, and find an abnormal slowdown of the longitudinal diffusion across the isotropic to nematic phase transition region. This phenomenon is counter-intuitive, since for other rod-like systems the longitudinal diffusion is expected to increase across the phase transition. We further compare the diffusion of F-actin, microtubule and fd virus in F-actin solutions across the transition region and find that the interaction between F-actin and microtubule or F-actin and fd virus are much smaller than that between actin filaments. Also, the microscopic viscosity probed by F-actin is found to increase sharply with [M g 2+ ] in the nematic but not in the isotropic state. We propose that the abnormal slowdown 127 is due to the transient associations between actin filaments mediated by divalent counterions, which is more favorable in the nematic phase. A simple model based on this picture quantitatively explains the dependence of microscopic viscosity on [M g 2+ ] and qualitatively accounts for the different behaviors among different probe filaments. This abnormal slowdown across transition region is consistent with the enhanced shear moduli in the nematic phase due to increased [M g 2+ ], since they are both caused by the counterion induced transient joints between parallel aligned filaments. In conclusion, a number of physical properties of F-actin as a reconstituted protein network have been better understood through this work. 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