BOSTON UNIVERSITY

COLLEGE OF ENGINEERING

Dissertation

PROBING WEAK SINGLE­MOLECULE INTERACTIONS:

DEVELOPMENT AND DEMONSTRATION OF A NEW INSTRUMENT

by

KENNETH ANDERS HALVORSEN

B.S., Clarkson University, 2000

Submitted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

2007

iii

ACKNOWLEDGEMENTS

I would like to thank Prof. Evan Evans for his continued support in all areas of my professional development. His mentorship has proved invaluable to the completion of this work. I would also like to thank Prof. Volkmar Heinrich for his scientific and technical support with the development and use of the instrument presented here. His insight, experience, and custom software greatly accelerated the development process.

Co­workers Wesley Wong and Koji Kinoshita have contributed countless thoughtful discussions that have shaped this work, along with countless more irreverent ones which have made the work more enjoyable. Long distance co­worker Andrew Leung has graciously tolerated and answered all of my questions regarding laboratory and chemistry protocols for the preparation of functionalized beads.

Many thanks to my committee members for serving on my committee and often going far beyond the expected duties: Jerome Mertz, Ed Damiano, Shyam Erramilli, Amit Meller. I would also like to extend my appreciation to all others who helped or supported me: Andy Golden, Wynter Duncanson, Roberto Rief, Marc Herant.

Most importantly, my deepest love and thanks to my family for their constant support and encouragement, especially my wife Liz who has always been at my side, and our infant son Jackson who has brought new happiness and inspiration.

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PROBING WEAK SINGLE­MOLECULE INTERACTIONS:

DEVELOPMENT AND DEMONSTRATION OF A NEW INSTRUMENT

(Order No. )

KENNETH ANDERS HALVORSEN

Boston University College of Engineering, 2007

Major Professor: Evan Evans, Professor of Biomedical Engineering

ABSTRACT

The focus of this thesis is the development, verification, and scientific use of a force probe designed to explore weak interactions between or within single biomolecules. The developed system utilizes a laser optical trap to apply piconewton forces to a functionalized probe bead that interacts with a reactive test bead, and high­speed video processing to measure nanometer displacements of both beads at a rate of ~1500 Hz. The position of the probe bead is used to report both the force on a single molecular bond and the change in length due to protein unfolding or refolding. Several feedback and automation systems have been integrated to facilitate the repetitive testing required to characterize these stochastic interactions.

The measurement and application of piconewton­sized forces depends on accurate position detection and proper calibration of the optical trap. The accuracy of the position detection was determined to be 2­3 nm by analyzing a non­moving bead. To assess optical trap calibration, a detailed study and comparison of common methods was performed. Noise­based methods which rely on the variance or power spectrum were found to have systematic errors due to the finite integration time of the detection. To solve this problem, the effect of integration time on these measurements was quantified and new calibration methods were developed.

Finally, the instrument was used to study the forced unfolding of the spectrin repeat, a small modular protein domain found in various load supporting proteins. The unfolding kinetics were determined by pulling on the molecule over several orders of magnitude in loading rate. The findings indicate that spectrin unfolding is governed by a single prominent energy barrier, characterized by a force scale of 2.5 pN and a stress­free unfolding rate of 0.003 s ­1 .

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Table of Contents

Title Page.............................................................................................................i

Readers Approval Page..................................................................................... ii

Acknowledgements........................................................................................... iii

Abstract .............................................................................................................iv

Table of Contents ...............................................................................................v

CHAPTER 1: INTRODUCTION ..............................................................................1

1.1. Motivation...............................................................................................1

1.2. Goals .......................................................................................................2

CHAPTER 2: BACKGROUND.................................................................................4

2.1. Optical Trapping ....................................................................................4

2.2. Chemical Kinetics...................................................................................7

CHAPTER 3: INSTRUMENT DEVELOPMENT .............................11

3.1. Principle of Operation..........................................................................11

3.2. Instrument Design ................................................................................12

3.3. Laser Alignment ...................................................................................18

3.4. Particle Tracking..................................................................................19

3.5. Optical Trap Calibration .....................................................................22

3.6. Force Application/Micropipette Control .............................................22

3.7. Micropipette Feedback.........................................................................23

3.8. Bead Surface Chemistry.......................................................................24

3.9. Experimental Procedures.....................................................................25

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CHAPTER 4: OPTICAL TRAP CALIBRATION I: COMPARISON OF STANDARD METHODS.................................................................30

4.1. Common Calibration Methods ............................................................30

4.2. Extent of the Harmonic Regime...........................................................41

4.3. Other Considerations ...........................................................................42

CHAPTER 5: OPTICAL TRAP CALIBRATION II: CALIBRATING WITH IMAGE BLUR......................................................................45

5.1. Abstract.................................................................................................45

5.2. Introduction..........................................................................................45

5.3. Bias in the variance of a harmonically trapped Brownian particle ..................................................................................................46

5.4. Numerical Studies.................................................................................48

5.5. Experimental verification.....................................................................49

5.6. Discussion: Practical suggestions for calibrating an optical trap .......53

5.7. Conclusions ...........................................................................................54

5.8. Appendix A: Calculation of the measured variance of a harmonically trapped Brownian particle ............................................55

5.9. Appendix B: High­pass filtering in variance measurements...............60

5.10. Appendix C: Experimental power spectrum calibration ....................61

5.11. Appendix D: Approximate analytical expression for k.......................63

CHAPTER 6: FORCED UNFOLDING OF THE SPECTRIN REPEAT..............64

6.1. Abstract.................................................................................................64

6.2. Introduction..........................................................................................64

6.3. Materials and Methods.........................................................................65

6.4. Results...................................................................................................68

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6.5. Discussion..............................................................................................73

6.6. Conclusions ...........................................................................................75

CHAPTER 7: FUTURE WORK: IMPROVING INSTRUMENT OPERATION AND THROUGHPUT..............................................76

7.1. Overview ...............................................................................................76

7.2. Experiment Preparation.......................................................................76

7.3. Instrument Use .....................................................................................80

7.4. Data Analysis ........................................................................................82

7.5. Other Considerations ...........................................................................83

7.6. Summary...............................................................................................83

CHAPTER 8: APPENDICES ..................................................................................84

8.1. Appendix A: Chemical Kinetics...........................................................84

8.2. Appendix B: Instrument Setup Guide .................................................86

BIBLIOGRAPHY........................................................................................................88

CURRICULUM VITAE..............................................................................................96

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Chapter 1: Introduction

1.1. Motivation The direct study of single molecule interactions is still a relatively young field, yet it has already provided groundbreaking new insight into a large number of biomolecular interactions (e.g. dynamic strength of biotin­streptavidin [Merkel et. al., 1999], unfolding of titin domains [Rief et. al., 1997; Oberhauser et. al., 2001], unfolding and refolding of RNA hairpins [Liphardt et. al., 2001], unfolding and refolding of ubiquitin [Fernandez et. al., 2004], and myosin stepping kinetics [Rief et. al., 2000]). These studies are made possible by sensitive force probes with the requisite sub­piconewton to nanonewton force resolution, and it is not surprising that developments in the field are often closely tied to technological advancements in force probes.

Several different force­probe instruments have been utilized to perform these single molecule measurements including the atomic force microscope (AFM) [Binnig et. al., 1986; Florin et. al., 1994; Carrion­Vazquez et. al., 1999], the optical trap [Ashkin, 1970; Ashkin et. al., 1986] , the biomembrane force probe (BFP) [Evans et. al., 1995; Simson et. al., 1998] and magnetic pullers (sometimes called “magnetic tweezers”) [Ziemann et. al., 1994; Heinrich et. al., 1996]. These force probes all act as linear springs (to some approximation) to pull on single molecules with well defined forces. Typically, single biomolecules are immobilized on a larger surface such as a bead, cover slip, or cantilever and their interactions are revealed by the nanoscale displacements of these larger surfaces.

To appreciate the benefits of single molecule experimentation, it is important to recognize that biomolecular interactions are generally weak (e.g. ~5­30 kBT), under stress, and far from equilibrium. A consequence of weak interactions is that Brownian impulses from the environment play a significant role in the mediation of molecular transitions. With this understanding, mechanical strength must be treated as a stochastic quantity, given by the probability of failure within a certain time. These interactions can be dramatically affected by force [Evans et al., 1997; 2002]. With this in mind, it becomes clear that a complete picture of a biomolecular interaction requires understanding the relationship between force, time and chemistry – a task well suited for single molecule research [Evans, 2001].

The distinction of single molecule research from other bulk chemistry methods lies in the ability to see beyond ensemble averages that can hide important features of molecular interactions. With single molecule experiments, it is possible to understand biological interactions at a molecular level, and to incorporate this understanding into larger scale mesoscopic behavior. By directly imaging the dynamic behavior of molecules under

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force, it is possible to study interesting biomolecular features such as forced unfolding and refolding of protein domains, mechano­sensitive switches, and multiple unbinding pathways.

1.2. Goals The two specific goals of this project were: 1) to develop and characterize a new single­ molecule research tool and, 2) to demonstrate its effectiveness with a study of the forced unfolding of the spectrin repeat, a small protein domain.

1.2.1. Specific Aim 1: Instrument Development The instrument was designed to characterize interactions between and within single biomolecules, and its specifications were dictated by the nature of these weak interactions. Specifically, the new instrument is capable of applying and measuring piconewton forces, measuring nanometer displacements, and performing a large number of tests in a short time, with the added benefit of minimal user intervention.

The system utilizes a laser optical trap to apply piconewton forces to a functionalized probe bead that interacts with a reactive test bead, and high­speed video processing to measure nanometer displacements of both beads at a rate of ~1500 Hz. A single test is performed by bringing the test bead into feedback­controlled contact with the probe bead and subsequently retracting the test bead. The position of the probe bead is used to report both the force on a single molecular bond and the change in length due to protein unfolding or refolding. However, due to the stochastic nature of these interactions it is not sufficient to perform a single test. Instead, a large number of statistics must be collected to characterize each interaction, highlighting the need for automated testing.

This new system offers the following features: § Soft, tunable spring with a wide range of spring constants

­ Spring constant range from 0.001 pN/nm to 0.1 pN/nm § High spatial and temporal resolution

­ 2 nm resolution at 1500 Hz § Precisely controlled application of force

­ Forces as low as several femtonewtons § Programmable touch force

­ Real­time tracking with feedback for well defined and repeatable touches § No inherent limitations on the local environment

­ Testing can be performed in a variety of media § Automated software control

­ Automated control yields up to 1000 tests per hour with minimal supervision § Drift stabilization

­ Feedback on the pipette eliminates alignment problems from slow drift

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1.2.2. Specific Aim 2: Forced unfolding of the Spectrin repeat To demonstrate its scientific potential, this instrument was employed to study the mechanical stability of the spectrin repeat, a three helix protein domain often found in structures subject to large mechanical stresses (e.g. red blood cell cytoskeleton, muscle sarcomere). While a few other groups have successfully measured the forced unfolding of spectrin repeats using AFM [Law, et. al., 2003; Rief et. al., 1999; Altmann et. al., 2002], little work has been done to quantify the unfolding kinetics.

The work presented here captures a detailed view of the unfolding dynamics of the spectrin repeat for the first time. An engineered spectrin construct was pulled end to end with the new instrument to measure both the force of unfolding events as well as the length gained from unfolding. By varying the force loading rate over several orders of magnitude, the force dependence of spectrin unfolding was characterized with a simple model. The extrapolation to zero force yielded an off­rate of 0.003 s ­1 , in close agreement with off­rate measurements performed in solution [Scott et. al., 2004].

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Chapter 2: Background

2.1. Optical Trapping

The optical trap is a relatively recent breakthrough, allowing the 3D confinement of small particles without mechanical contact [Ashkin, 1970]. The premise is to exploit the force of radiation pressure, which arises due to momentum of light itself. For typical incoherent light sources, the forces from radiation pressure are too small to be useful for micromanipulation, but the advent of lasers has fostered forces large enough to accelerate, guide, or trap small particles.

An optical trap is formed when laser light is focused through an objective with high numerical aperture, exchanging momentum with polarizable materials such that the material is “pulled” toward the focus. For small displacements from the focus, the optical trap acts like a linear spring with a spring constant proportional to the laser intensity. Thus, an optical trap provides a soft and tunable spring, perfectly fit for exploring single molecule interactions and transitions.

There are two main regimes useful for describing the physics of optical trapping: ray optics and Rayleigh scattering. Each of these regimes is valid for a range of particle sizes. Ray optics is valid for particles much larger than the wavelength of light while Rayleigh scattering is valid for particles much smaller than the wavelength of light. Unfortunately, many optical trapping experiments utilize beads whose dimension is similar to the wavelength of light where neither ray optics nor Rayleigh scattering apply. Calculation of the optical forces for an arbitrary sized bead is nontrivial, requiring the calculation of the electromagnetic field with appropriate boundary conditions. The Lorenz­Mie theory [Mie, 1908] solved the problem of a dielectric material in a plane wave, but it cannot describe the commonly used Gaussian beam. For a Gaussian beam, the Generalized Lorenz­Mie Theory [Gouesbet et. al., 1988; Gouesbet, 1999] can be used to calculate optical forces with reasonably accurate numerical results [Nahmias and Odde, 2002].

While each regime provides insight into the workings of an optical trap, they are generally not used to quantitatively describe an optical trap. Instead, optical traps are typically calibrated prior to use, providing a more accurate measure of the trapping force as a function of position. Several common methods exist to calibrate optical traps, and these are explored in detail in Chapter 4: Optical Trap Calibration.

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2.1.1. Ray Optics Regime When the particle size is bigger than the wavelength of the light, the ray optics approximation can prove useful in understanding trapping forces [Ashkin, 1992]. By tracing light rays interacting with a sphere and following the transfer of momentum, the physical basis for the optical trap becomes clearer (see Figure 1).

Figure 1 – Momentum transfer ∆p of a single ray as light refracts through a sphere of refractive index n2 from a medium of index n1 (a), and momentum transfer of two rays of different intensity (b) – illustrating why particles are pulled toward the highest intensity

[http://www.uni­leipzig.de/~pwm/kas/modul_opticalforces/theory.html].

2.1.2. Rayleigh Scattering Regime For particle sizes much smaller than the wavelength of light, the particle can simply be treated as an induced dipole due to the polarization of dielectric material in an electric field. The force on such a dipole divides into two terms: a scattering force in the direction of the incident light, and a gradient force in the direction of the intensity gradient of the light. The origin of the gradient force is not always obvious, but can by derived simply. If d is the induced dipole of a dielectric sphere, then:

d E = α ur ur

(2.1)

where α is the polarizability. The energy of a dipole in an electric field is:

2 U d E E = − • = −α ur ur

(2.2)

The average force is then:

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2

0 2 F U E I

c α

= −∇ = α∇ = ∇ ε

ur (2.3)

2.1.3. Applications The optical trap has already proved to be an invaluable tool in biology, with applications such as micromanipulation [Grier, 2003], cell surgery [Ashkin and Dziedzic, 1989; Berns et. al. 1998], and sorting [Wright et. al., 1998; Applegate et. al., 2004]. In the single molecule field, optical traps have been used to provide significant insight into the mechanical properties a variety of molecules. Notable examples include titin unfolding and refolding [Kellermayer et. al., 1997], movement of kinesin [Svoboda et. al. 1993; Asbury et. al. 2003], and force extension properties of DNA [Wang et. al., 1997].

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2.2. Chemical Kinetics

2.2.1. Biomolecular Interactions Non­covalent interactions between and within biomolecules (e.g. receptor­ligand pairing or protein unfolding) are typically weak (5­30 kBT) when compared with the covalent bonds responsible for holding the molecules together (100 kBT or more). Consequently, Brownian impulses from the aqueous environment play a significant role in the mediation of biomolecular transitions, facilitating complex biological processes without large energetic requirements. Thermal activation alone is capable of disrupting these weak interactions on timescales ranging from microseconds to a year.

A folded protein or a receptor­ligand complex already represents a complicated mesoscopic system, composed of thousands to millions of atoms, and held together by at least several weak interactions such as hydrogen bonds. Despite these complexities, the energetics of biomolecular interactions is often well approximated by a simplified model of two distinct chemical states with different energies (representing bound and unbound or folded and unfolded) and an intervening barrier. Transitions between states on this energy landscape are dictated by forward and reverse reaction rates, koff and kon (Figure 2).

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Figure 2 – A one­dimensional energy landscape portraying a two state system (bound/unbound) separated by a single energy barrier (dashed line). The addition of force adds a linear potential, tilting the landscape and lowering the barrier (solid line)

[Evans and Williams, 2002].

2.2.2. The role of force The application of force to pull a bond or structure adds a linear mechanical potential to the energy landscape (Figure 2). This mechanical potential is proportional to the projection of the force along the spatial pathway, and has the effect of tilting the landscape, thereby lowering, shifting and narrowing the energy barrier [Evans and Ritchie, 1997]. The lowering of the energy barrier has the profound effect of dramatically increasing the rate of forward transition, koff, while lowering the reverse rate, kon. In the simplest case where the energy landscape is comprised of two narrowly bound states separated by a single sharp barrier, the transition rate is exponentially dependent on force [Bell, 1978].

( ) ( ) exp off off f t k t k f β

≈

o (2.4)

Where k o off is the unstressed off rate and fβ is the force scale for the interaction given by:

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B k

ts

T f x β = (2.5)

Where xts is the projected location of the energy barrier along the reaction coordinate.

2.2.3. Dynamic Force Spectroscopy The development of dynamic force spectroscopy (DFS) has provided a useful framework for testing single­molecule transitions with external fields [Evans and Ritchie, 1997; Evans, 2001; Evans and Williams, 2002]. It has been successfully used to probe a number of biomolecular interactions [Rief et. al., 1997; Fritz et. al., 1998; Merkel et. al., 1999; Evans et. al., 2001]. The crucial concept is that force­mediated transitions in weakly interacting systems do not occur at a single characteristic value of the applied force. Instead, thermal activation couples force to time, requiring a spectrum of force loading rates to characterize an interaction.

For a linear force ramp, the single barrier model predicts that the most likely breaking force is proportional to the logarithm of the loading rate:

* ln f

off

r f f

f k β β

=

o (2.6)

Where f * is the most likely breaking force, rf is the force loading rate, k o off is the unstressed off­rate, and fβ is the force scale given by equation (2.5). A full derivation is shown in Appendix A: Chemical Kinetics.

2.2.4. Performing DFS Experiments As an example of how to apply dynamic force spectroscopy to a molecular interaction, consider an interaction between a receptor A and ligand B. A single associated A­B complex is mechanically pulled apart with a linear ramp in force until the complex dissociates. For each pull, the dissociation force is recorded with the loading rate. Since dissociation is a stochastic process, the single test needs to be repeated several times (preferably hundreds) to build a histogram of unfolding forces at a single loading rate. With enough statistics, the shape of the histogram is predicted by Equation (8.10).

The entire process is then repeated for other loading rates, ending with a set of histograms portraying the probability densities of dissociation for each loading rate. The most likely unbinding force occurs at the peak of each histogram, and a plot of the most likely unbinding force vs. the log of the loading rate should yield a straight line according to Equation (2.6). Performing a least squares fit of the data with Equation (2.6) yields the

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spontaneous off­rate as well as the force scale, which completely characterize an ideal single barrier interaction.

The simple theory described here is based on an idealized interaction consisting of a single sharp barrier in the energy landscape. However, real energy landscapes could assume a variety of shapes, and need not consist of even a single barrier. The theory can be expanded to accommodate the higher complexity. The model also assumes that the interaction is far from equilibrium, meaning that once dissociation has occurred there is practically no chance for reassociation. For typical loading rates, this turns out to be a good approximation, but slower rates may have a nonzero rate of reassociation. In these cases, a plot of most likely breaking force vs. log loading rate may be linear for some portion, but have curvature at slower rates as reassociation plays a role.

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Chapter 3: Instrument Development

3.1. Principle of Operation The force probe instrument presented here is based on a relatively simple idea: allow two single molecules to interact, apply a known force to pull them apart, and quantify the conditions of intermolecular and/or intramolecular transitions.

The instrument utilizes a functionalized probe bead to report the formation and failure of single biomolecular bonds as well as intramolecular changes such as protein unfolding and refolding. The probe bead is held by an optical trap, which acts as a tunable linear spring for small displacements. Another functionalized glass bead (“test bead”) is rigidly held by a micropipette mounted to a piezo translator. The translator moves the test bead into contact with the probe bead, and retracts upon real­time detection of the touch. The formation of a bond, as well as the force it experiences, is reported by the displacement of the probe bead from the trap center. Intramolecular transitions are revealed by abrupt changes in distance between the two beads.

The instrument has two modes of operation: force ramp mode and force clamp mode. In both cases, the test bead is moved toward the probe bead at a constant approach rate, contacts the probe bead until a threshold touch force is reached, and holds the touch force for a set touch time. In force ramp mode, the test bead then moves away from the probe bead at a set force loading rate (i.e. constant speed) until reaching the end of travel. In force clamp mode, the test bead is pulled quickly to a set value of force and is held at that position until the user ends the cycle.

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Figure 3 illustrates the general principle of operation of the force probe. The schematic spring is drawn for visualization purposes only, to show the lateral (x­axis) direction of force application by the 3D optical trap utilized in this setup. The bead in the optical trap is denoted the “probe bead” and the bead in the micropipette is denoted the “test

bead”.

3.2. Instrument Design

3.2.1. Description The force probe instrument was constructed around an inverted light microscope (Zeiss Axiovert S100) atop a breadboard style optical table (Oriel 77G­491­02), allowing convenient and rigid attachment of optical components. A detailed layout of major components is shown in Figure 4 and a digital image of the instrument is shown in Figure 5. Briefly, the force probe consists of an optical trap, an imaging system, a motion control system, and a computer interface. These will be discussed in detail in the following sections as well as other design considerations such as noise isolation and safety. To avoid confusion, subsequent sections will refer to the axis along the micropipette as the x­axis, the optical axis will be the z­axis, and the other will be the y­ axis, as shown in Figure 3.

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Figure 4 ­ shows a diagram of the main components of the system. The labeled parts are: (a) halogen lamp, (b) test chamber, (c) piezo translator with rigidly mounted

micropipette, (d) three­dimensional translator, (e) microscope stage, (f) microscope objective, (g) laser head, (h) laser power controller, (j) laser beam expander, (k) Sensicam high­speed CCD camera, (m) computer. Dashed lines indicate computer

interfaces to hardware components.

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Figure 5 – Digital image of the force probe instrument. The laser and laser optics (not pictured) are behind the microscope and to the right along the optical table.

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3.2.2. Optical trap The optical trap is formed by focusing laser light through a high numerical aperture objective (see section 2.1). The laser and supporting optics are discussed in this section.

Laser The laser is a 4 Watt continuous wave laser (Coherent Compass 1064­4000M) with a wavelength of 1064nm. This wavelength was chosen due to its low absorption in water. Other common laser wavelengths have the disadvantage of causing significant local heating near the focus, or causing damage to biological materials. The local heating for a small (500 nm) optically trapped bead has been measured experimentally to be about 8 K/W [Peterman, et. al, 2003], decreasing with increasing bead diameter. Based on this information, a typical bead in our trap (2 µm bead with 250 mW laser power) should have local heating of less than 2 K. This laser was also chosen for its high pointing stability (< ±5 µrad), which is of particular importance when measuring on the scale of nanometers.

Laser Optics A variety of optical components are required to aim the laser in the right direction and to control the beam characteristics such as intensity and width. These components will be discussed briefly, beginning with those closest to the laser. Immediately following the laser is an LCD based laser power controller (Brockton Electro­Optics Corp. LPC­NIR), which modulates and stabilizes the laser intensity. The laser intensity can be controlled on the unit or via computer control.

A high speed shutter (Uniblitz LS62M2) and controller (Uniblitz VMM­D3) are used to toggle the optical trap on or off, since it is often inconvenient to have the optical trap on at all times. Next, the laser beam width is increased using a 5X beam expander (OFR, ELQ­25­5X­YAG) with fine control of beam divergence. This unit is mounted on a 2­ axis tilt stage (Newport ULM­Tilt) and a 3­axis translation stage (Newport 461 Series). This level of adjustability was important for simplifying the alignment procedure. Additionally, small adjustments in the tilt translate the optical trap position in the image plane.

Beyond the beam expander, a beam steerer (Newport BSD­2A) with two specially coated mirrors (Coherent 45­6004­000) directs the beam into the rear port of the microscope. Inside the microscope, the beam is reflected upward to the objective with an additional mirror. The laser is focused by the oil immersion objective (Zeiss plan neofluar 100x/1.3), which has been reported to have a transmissivity of 59% +/­ 2% at our laser wavelength [Svoboda and Block, 1994].

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3.2.3. Imaging The imaging system creates transmitted light images at the focus and captures them with 2 different cameras. The various components in the imaging system will be discussed starting with the illumination source.

A 100W halogen light source (Zeiss HAL 100) passes white light through the condenser lens, where it is focused into the test chamber. The same objective used for the optical trap (Zeiss plan neofluar 100x/1.3) collects the light and images are formed on 2 different cameras after passing through the internal optics of the microscope.

Two cameras serve the different purposes of performing high speed bead tracking and general imaging. The high speed camera (Cooke Sensicam High Performance) is connected to the top camera port via a camera adapter (Zeiss 456140) and a 4x TV tube (Zeiss 452985). It is capable of recording single­line images at a rate of 1500/sec and has a final magnification of 24nm/pixel. The general use camera (MTI CCD­300T­RC) is a standard monochromatic camera with video rate capabilities, which is connected to the side camera port. The general use camera is needed since use of the microscope eyepieces is unsafe when the laser is on. This camera is used for laser alignment, picking up beads with the micropipette or optical trap, focusing, verifying proper alignment of the two beads, and performing automated micropipette feedback (see section 3.7).

3.2.4. Motion Control Both coarse (µm­mm size) and fine (nm size) adjustments are available to position the microscope chamber, micropipette and objective lens. The original stage of the microscope was replaced with a custom designed stage with a gridded one inch hole pattern for fastening components and a central aperture for the objective.

Chamber Positioning A two­dimensional translating stage with central aperture (Newport 426 series) allows x and y positioning of the microscope chamber. The position of the stage is controlled by motorized micrometers (Oriel motor mike). These micrometers have been interfaced with a computer joystick (Microsoft Sidewinder Precision 2) via custom software for convenience.

Micropipette Positioning The micropipette assembly is mounted directly to a closed loop 1D piezo translator (Physik Instrumente P­753.12C), which controls the x­position of the micropipette with sub­nm accuracy. The piezo amplifier (Physik Instrumente E­505.00) and controller (Physik Instrumente E­509.C1A) allow piezo control either by hand or by computer interface. This 1D translator is rigidly mounted to a 3D translator (Newport 461 series) with hand micrometers (Newport HR­13) for coarse positioning by hand. The 3D translator has been modified to incorporate an additional 1D piezo translator (Physik Instrumente P­170.07) to control the y­position with nm accuracy. This high precision y­

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positioning is useful for adjusting bead alignment at the start of an experiment or to compensate for slow microscopic drifts. This 1D translator uses a high voltage amplifier/controller (Physik Instrumente E­507.00), again allowing either hand or computer control.

Objective Positioning The position of the objective (and thus the focus) can be adjusted coarsely by using the standard focus knob on the microscope. Additionally, a 1D piezo translator (Physik Instrumente P­721.10) is mounted below the objective for sub­nm adjustment of the focus. It utilizes an amplifier (Physik Instrumente E­505.00) and a servo controller (Physik Instrumente E­509.X1), allowing control by hand or computer. This precise positioning of the objective translates the optical trap along the z­axis with sub­nm precision. This becomes useful for performing experiments at a known distance from the cover slip as well as compensating for slow microscopic drifts.

3.2.5. Computer Interface Interfacing equipment with computers was an important part of instrument development, since a main goal of the project was to minimize user interaction. Most of the motion control equipment can be controlled through an external analog signal, which was generated by an analog output card (National Instruments PCI­6733) or a wave generator (American Avantech Corp. PCL­726). These cards were controlled by a variety of custom software designed with standard programming languages (National Instruments Labview 7.0, C++).

Both the high speed and general use camera were connected to computers via a computer control card (Cooke) and a framegrabber (Matrox Meteor II), respectively. The high speed camera and general use camera are computer controlled via custom software programmed in C++ and labview, respectively.

3.2.6. Noise Isolation Upon preliminary testing of the device, large fluctuations in position were recorded for trapped beads. These fluctuations were determined to be largely due to air currents in the room. To minimize these currents and other mechanical noise, an enclosure was built to isolate the instrument from the surrounding space. This enclosure dramatically reduced these large fluctuations.

The enclosure was custom designed and assembled from pre­cut aluminum and lexan pieces (80/20 Inc.) and fits snugly around the microscope and beam steerer. Sliding doors on both sides allow hand manipulation inside the enclosure for tasks such as micrometer adjustment and focusing. A rear aperture lets the laser light in without obstruction.

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3.2.7. Safety Considerations Extreme caution must be exercised when working with this powerful class 4b laser. Severe eye damage or blindness can result from direct or indirect (e.g. reflections) exposure to the invisible laser light. As a result, this instrument was built in an isolated room and an illuminated sign was added to the door to indicate when the laser is in use. Laser goggles specifically designed for the near­IR wavelengths (LaserVision L868) were worn at all times when the laser is on. Additionally, the lab worked closely with the department of Enviromental Health and Safety to work toward compliance of BU’s laser safety policies.

3.3. Laser Alignment Proper alignment of the laser is critical to the operation of this instrument. Typically, laser alignment only needs to be performed upon initial construction and changes in optical components. However, it is useful to periodically check the alignment as equipment can settle or drift over time, either naturally or due to accidental human contact.

Alignment of the optical trap was performed at low power (~10 mW) using an IR sensing card (Lumitek IR sensor screen), an IR viewer (Electrophysics ElectroViewer 7215), and the two cameras. It is important to align at low power to minimize risk of eye injury and damage to optical components.

First, a rough alignment of the optical components and the laser was performed using the IR card and the IR viewer. This was done in the absence of focusing elements, to establish the laser path. Once the optical components were roughly aligned, the MTI camera was used to center and fine­tune the optical trap (this can be done since this camera has near­IR sensitivity).

Next the microscope objective was added and a slide with water was placed on the stage. With the laser is turned on, a small portion of the IR laser light is reflected at the surface of the glass­water interface. A portion of this light is diverted to the MTI camera, so the laser can be seen on a monitor if it is passing through the objective. To center the laser with the objective, the beam steerer was adjusted until the laser spot could be seen at the center of the camera with maximum brightness.

To ensure that the laser path and the imaging path are coincident, the microscope focus can be scanned through the region that includes the laser focus. With an ideal alignment, the laser spot should be radially symmetric at all depths of focus. Any deviation from the ideal case can be corrected by adjusting the two mirrors.

19

Once this alignment has been established with the objective in place, other optical components such as the beam expander can be inserted into the laser pathway. The laser can be re­aligned by adjusting the translation and tilt of the newly added component and again verifying alignment with the camera.

Finally, the optical trap needs to be adjusted so that the image focus and the optical trap are in the same plane. This can be accomplished by adjusting the divergence of the beam expander, which translates the optical trap in the z­direction.

3.4. Particle Tracking

3.4.1. Method Using techniques and software previously developed in our lab, 1D displacements of small (micron size) beads are measured to within ~2nm, at a rate of 1500 Hz. This is accomplished using custom real­time tracking software with video microscopy images supplied by the fast Sensicam camera. To operate at the maximum frame rate of 1500 frames/second, it is necessary to limit the image to a thin strip of only a few lines (up to 32), which are binned by the camera into a single line. When the center of this thin strip coincides with the axis of symmetry of the image, the single line output by the camera contains all the data necessary to track these spherical particles. Since there is nearly no variation in the y­direction within the thin strip (i.e. all lines are nearly identical), the binning acts to increase the signal to noise ratio. The single line is output as an intensity profile to the custom tracking software. A microscope image of a bead along with its intensity profile is shown in Figure 6.

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Figure 6 – A microscope image of the two beads (above). The probe bead (left) is confined by the optical trap and the test bead (right) is held by suction in the

micropipette. The intensity profile of the horizontal center line (below) shows a dramatic landscape representing regions of extreme brightness and darkness. The peaks and

valleys of this landscape are tracked to give high accuracy 1D tracking.

In the ideal case, the experiments are performed in a single plane of focus with motion in only the x­direction of that plane. Thus, images of the beads are constant throughout the experiment, only being translated in one dimension. Consequently, the intensity profile of the beads will retain their shape but translate their position as the beads are moved. The tracking software allows the user to identify regions along the axis of motion where a bead edge will be found. The software then “locks on” to this edge by fitting several points near an extreme with a polynomial fit. The position of the extreme of the polynomial fit is then used to determine the position of the bead with sub­pixel accuracy. The software is able to track multiple beads in this way.

3.4.2. Accuracy The accuracy of position detection was tested by tracking the position of opposite edges of a bead immobilized on a cover slip. The true difference of these positions is a fixed length, but the measured distance fluctuated slightly in time due to measurement error, yielding a Gaussian distribution about a mean value (Figure 7). The standard deviation of this distribution yields an estimate of the instrument error in position, given by:

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distance position 2

σ σ = (3.1)

Where σposition is the standard deviation of the position measurement and σdistance is the standard deviation of the of the distance measurement. For the data shown below, σdistance

is 2.7 nm, yielding σposition of just less than 2 nm.

Figure 7 – The measured distance between two edges of an immobilized bead relative to the average distance. Since the actual distance is constant in time, the time trace (top) and histogram (bottom) demonstrate the error in the particle tracking. Fitting the histogram with a Gaussian gives an estimate of ±2 nm for tracking accuracy.

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3.5. Optical Trap Calibration Calibration of the optical trap is necessary to determine the force on a bead at a given position. A complete calibration provides a way to convert positions (which we measure) to forces in any region of the trap. A simplified calibration provides a spring constant which describes the trap behavior in the small region where it behaves linearly. Treating the optical trap as a linear spring simplifies both the calibration procedure and the application of a linear force ramp during experimentation. These calibration procedures are described in detail in Chapter 4: .

3.6. Force Application/Micropipette Control Accurate control of the micropipette (and thus the test bead) is crucial to applying force in a well known way (such as a linear ramp). Since the optical trap creates an energetic potential (harmonic for a small region) on the probe bead, moving this bead from the trap center causes a restoring force. Thus, application of force is dictated both by the optical trap itself and by the ability to accurately position the probe bead in space. Performing an experiment on a single molecule requires a molecular connection between the probe bead and test bead, allowing positioning of the test bead to effectively control the force application.

As mentioned in section 3.2.4, the micropipette position in the x­dimension is controlled by a piezo device with sub­nm accuracy. To verify the accuracy of the piezo, a test bead was tracked while the piezo was moved in a triangle wave at a set velocity (Figure 8). The result indicates that the set speed and the measured speed differ by less than 1%.

23

Figure 8 – Tracking a test bead in the micropipette while moving in a triangle wave at 30 µm/sec. Fitting a line to the slopes of one cycle yields near perfect agreement of 30.16

µm/sec and 30.11 µm/sec, an average error of less than 0.5%.

3.7. Micropipette Feedback Two independent feedback control systems were incorporated into the system to ensure repeatable conditions for each test. One feedback system operates in the x­direction, controlling the touch strength and touch time, while the other operates in the y and z directions, controlling the alignment of the two beads to compensate for slow drifts.

3.7.1. Controlled Touch and Retract Cycles The experimental process of conducting a large number of tests with beads has been automated with custom software. This software controls the motion of the test bead according to user input parameters. Before starting a force ramp experiment, the user must enter the touch and retract parameters: touch time, touch force, approach speed, retract speed, and spring constant.

Upon starting the experiment (after proper alignment), the test bead will move toward the probe bead at the set approach speed. The touch against the probe bead is detected by the software and feedback­controlled to insure a repeatable touch force for each cycle. After being held at the set touch force for the set touch time, the test bead is retracted at the set retract speed to finish the cycle. These repetitive cycles continue until the user stops the program. This level of automation is crucial because it allows the large number of tests to be performed in a relatively short time and with minimal user intervention. An analogous program exists for force clamp experiments, allowing keyboard control of the clamp time.

3.7.2. Automatic Alignment Consistent micropipette alignment throughout an experiment is important for ensuring high quality data. Slow microscopic drift can cause the micropipette to move in all three dimensions. Drift in the on­axis dimension is not a concern, since the software has the touch feedback described above.

Drift in the other two dimensions can cause misalignment. To deal with this problem, feedback control software was developed. The Labview based software uses the image from the MTI camera to constantly align the micropipette along the y­axis and ensure a consistent focus (i.e. alignment on the z­axis).

The software works by tracking a portion of the micropipette with an autocorrelation algorithm. A box is drawn in the region consisting of the two sides of the micropipette

24

and the edge of the test bead within the micropipette. When the tracking is activated, this box moves with the micropipette by comparing the initial image upon activation and the current image. The position of this box in the y­dimension can be controlled by entering the desired position and activating the y­axis feedback. If the current position is not the same as the desired position, a piezo translator moves the micropipette in small increments until it is within a set tolerance of the desired value.

Feedback for the z­axis is controlled by measuring the variance inside the box. The variance changes dramatically with focus, and setting a value for desired variance refocuses and holds the focus at the desired value by moving the z­axis piezo translator.

To ensure that the experiment is not disrupted by motion due to feedback, the y and z axis motion is limited to occur beyond a threshold on the x­axis. This allows feedback motion only when the test bead is not in contact with the probe bead.

3.8. Bead Surface Chemistry

3.8.1. Silanization Proteins and other biologically relevant molecules can be covalently attached to the surface of glass beads using various silane chemistries. Several silanes are commercially available that covalently link a reactive group to a glass surface. Common examples are amino silanes and mercapto silanes which “activate” glass by decorating its surface with exposed amino groups and thiol groups, respectively. Following silanization, the beads can be dried and stored for up to 6 months.

3.8.2. Protein Linkage Once the glass has been “activated” with silane, a bifunctional linker molecule is often used to attach the protein to the glass. The linker also serves as a molecular spacer, keeping the protein several nanometers from the surface to minimize short­range nonspecific interactions between the surfaces. The linker I most commonly use is maleimide­PEG3400­NHS. Maleimide forms a covalent link with mercapto­silanized glass whereas the NHS covalently attaches to an exposed amine group of the protein. This method has been summarized previously [Evans et. al., 2001].

One of the main challenges with immobilizing biomolecules on glass surfaces is linking the glass to a known region of the molecule. Often the chemical linkage specifically binds to a chemical group which may be present in large numbers (as amines often are), leading to molecules which are bound to the surface in slightly different ways. For instance, if a molecule has one amine at each end, there may be two populations of the molecule with different ends sticking away from the bead surface.

25

This uncertainty may not always be a problem, but there are a variety of ways it can be averted if it is. One way is to form a covalent reaction between the silane and something that has only one occurrence in the molecule of interest. For example, if a protein has only one amine group, then we can be reasonably certain that an NHS will react only with that amine. Typically, this is difficult with amines since they are so common, but cysteines are much rarer and can be linked specifically with maleimide.

Another way is to covalently immobilize antibodies to the surface and have the antigen present on the molecule of interest. While the antibodies on the surface will likely have some population of orientations, those with the binding pocket exposed will be able to bind with the antigen.

3.9. Experimental Procedures Conducting a single molecule experiment from start to finish involves several steps. These steps can be roughly divided into three categories, which will be described in this section: preparation, data collection, and data analysis.

3.9.1. Chamber Preparation Microscope chambers for experiments were custom made from thin (~1.5 mm) lexan by making a U shaped extrusion. Cover glasses were secured to both sides using vacuum grease, leaving one edge of the chamber open for insertion of the micropipette (Figure 9). The chamber is filled first with solution and then with beads. For ease of use, concentrated beads are carefully deposited in thin lines on either side of the chamber using a pipet. This physically separates the two bead populations by over 10mm, effectively isolating them for easy recognition. This also leaves a large bead­free space in the middle of the chamber, where experiments can be conducted without interference.

26

Figure 9 – 3D representation of the microscope chamber and micropipette. The vertical cone represents the laser light forming the optical trap in the chamber. The micropipette

is attached to the 1­D piezo translator and enters from the right side.

3.9.2. Micropipette Preparation Micropipettes were formed from 0.7­1.0 mm glass capillary tubes (Kimble 46485), using a micropipette puller (David Kopf Instruments Model 730). The micropipette puller heats the glass tube while pulling the two ends apart until rupture, creating two micropipettes with a long taper with a very fine tip (<1 µm). These micropipettes were further processed with a microforge (Narishige MF­900) to produce a small (~2­3 µm), flat tip suitable for partial aspiration of the glass beads. The micropipettes were filled with water using a micropipette filler (World Precision Instruments MF34G). The filled micropipettes were held by a custom designed chuck (Research Instrument), which has a side port for applying pressure and an end which connects with a set screw to the x­ direction piezo. For these experiments, the side port was connected to a 10cc syringe with flexible tubing to easily apply positive and negative pressure.

3.9.3. Instrument Use Once the chamber and micropipette have been prepared according to the above procedures, the experiment can be set up and run as outlined in Appendix 8.2.

3.9.4. Data Collection Once the experiment is started by pressing the “go” button, the data collection is automatic. The program records time, piezo voltage, piezo state, position 1 and position 2 (if used) continuously until “stop” is pressed and the experiment is terminated. Figure 10 shows typical force vs. time graphs obtained from the instrument for a single cycle in force ramp mode and force clamp mode, respectively. These graphs represent only a small portion of typically large data files that may continue for hundreds or even thousands of cycles, where time is counted from the start of the experiment.

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Figure 10 – The two modes of operation are force ramp (top) and force clamp (bottom). In force ramp mode, the force of an event at a given linear rate is recorded. In this case, a 100 pN/s force ramp causes bond rupture at 25 pN. In force clamp mode, the lifetime of an event at a given constant force is recorded. In this case, a 6 pN force causes bond

rupture in ¾ of a second.

3.9.5. Data Analysis The software that runs the experiment outputs raw data as explained previously. To analyze the large volume of data, new software tools have been developed using Labview and Matlab.

28

Raw Data Analysis A Labview program was created to analyze each cycle of data (i.e. touch and retract), identify relevant parameters, and condense to a more manageable data file. The program begins by automatically dividing the large data file into individual cycles, beginning and ending with the start of inward motion of the micropipette. Graphs of force vs. time and distance vs. time are shown (Figure 11), and the user moves cursors to identify locations where transitions occur (either unfolding or breaking). The program records a single line of data for each segment (i.e. a cycle with 4 unfolding events would have 5 segments) consisting of times, forces, slope, and cycle identifier.

Figure 11 – Screen shot of the Labview based raw data analysis program, with a force vs. time graph above and a length vs. time graph below. The user identifies each segment

with the cursors, which record relevant data.

Post processing Once the raw data has been processed with the labview program, the data is further processed with a Matlab program that combines data from multiple tests, calculates

29

unfolding distances, and reorganizes the data into an easier to use form. The final file created from the Matlab program contains all data required for analysis. Further sorting by criteria such as date, force and unfolding length can be performed manually with a spreadsheet program or automatically with an additional Matlab program.

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Chapter 4: Optical Trap Calibration I: Comparison of Standard Methods

As discussed in section 2.1, an optical trap creates a mechanical potential on small particles, acting to confine them near the laser focus. For spherical particles such as glass beads, the potential is approximately harmonic near the trap center, causing a linear spring­like behavior. To accurately quantify the potential (or at least the spring constant in the harmonic regime), it is typically necessary to perform a calibration.

4.1. Common Calibration Methods There are four common methods for calibrating optical traps: equipartition analysis [Florin et. al, 1998], power spectrum analysis [Gittes and Schmidt, 1998; Berg­Sorensen and Flyvbjerg, 2004], drag force analysis, and recoil analysis [Capitanio et. al, 2002]. The first two methods utilize thermal noise analysis to estimate the spring constant in the harmonic regime of the trap, while the other two are mechanical measurements which can image the entire trap potential.

Figure 12 – Free body diagram of a bead near the trap center (top) with schematic spring and dashpot in parallel (bottom). A restoring force fr acts toward the trap center,

31

and a friction force ff acts opposite of the bead velocity. Because of the small Reynolds number, inertia can be neglected. Typically, the only external force is the random

thermal force (not shown).

4.1.1. Equipartition Calibration Calibration using equipartition analysis is based on the simple idea that a harmonically confined particle will fluctuate in space due to thermal noise, and the magnitude of those fluctuations vary with the strength of confinement. For example, a weakly confined bead would experience larger fluctuations in position than a strongly confined bead. Specifically, the equipartition theorem predicts:

2 B

x

k T k = σ

(4.1)

where k is the spring constant, kB is the Boltzmann constant, T is the absolute temperature, and σx 2 is the variance of the position in the x­dimension. With this simple formula, calculation of the spring constant only requires knowledge of the temperature and the variance. The variance can be measured if the instantaneous bead position can be measured repeatedly. In practice, a fast position detection system can collect enough data to accurately calculate the variance in less than a minute.

A characteristic data trace of position vs. time for a trapped bead is shown below along with a histogram of these positions (Figure 13). The statistical variance can be calculated either directly from the data trace or from fitting the histogram with a Gaussian. For data with little or no drift, these values agree within one percent, indicating that the bead is indeed confined in a harmonic trapping potential (for small displacements).

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Figure 13 ­ A typical data trace of position vs. time for a trapped bead (top) and a histogram of the positions fit with a Gaussian to determine the variance (bottom).

The obvious benefit of this calibration method is the speed and ease with which it can typically be done. All it requires is an optical trap, a particle, and a detection method. However, there are several limitations to the method. First, the analysis assumes a harmonic potential, so only the harmonic region can be quantified. From the above example, the bead in a weak optical trap rarely experiences fluctuations greater than 100 nm, but the trap may produce a restoring force at 10 times that distance. This limits the range over which the optical trap can be calibrated. Second, low frequency microscopic drift can inflate the variance, causing an underestimate of the spring constant. One solution to this problem is discussed in Chapter 5: . Third, finite integration time of the detection system underestimates the variance, causing significant errors if not considered. This issue is discussed at length in section Chapter 5: .

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This method is very popular due to its speed and ease. Some of the drawbacks can be minimized or eliminated under certain conditions (i.e. very weak trap). However, it is important to note that this method assumes that the variance is comprised solely from the thermal motion of the bead. The variance will always be overestimated in the presence of other noise.

4.1.2. Power Spectrum Calibration The power spectrum calibration is closely related to the equipartition method in that it uses noise­based analysis to estimate a spring constant. However, this method utilizes the frequency content of the signal, fitting the power spectrum of position data with a theoretical model:

2 2 2 ( ) ( )

B

c

k T P f f f

= γπ +

(4.2)

where γ is the friction factor, f is the frequency in Hz, and fc is the corner frequency defined by:

2 c k f = πγ

(4.3)

where k is the spring constant. The model is based on the predicted power spectrum of a harmonically confined particle in an overdamped thermal environment. A complete derivation of this theoretical form is discussed in section Chapter 5: . The Lorentzian form of Equation (4.2) shows two distinct regimes. For frequencies f<<fc, the power spectrum is dominated by the particle confinement, with a constant value of 2 4 / B k T k γ . For frequencies f>>fc, the particle behaves like it is diffusing freely, with the power spectrum decreasing with 1/f 2 (Figure 14).

Practically speaking, this method of calibration includes a few more steps than the equipartition method. Starting with equally spaced position samples (as in Figure 13), the data is blocked into equal sized non­overlapping segments. The power spectrum is calculated for each segment and averaged across all segments (Figure 14). The purpose of blocking the data first is to effectively smooth the power spectrum, making the fit faster and easier to verify qualitatively. This data can then be fit with least squares fitting to Equation (4.2), with two free parameters yielding the spring constant, k, and the diffusion coefficient, D=kBT/γ. If the diffusion coefficient is already known, then the fit can be restricted using the known value or the two­parameter fit can be used to examine the fitting accuracy.

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Figure 14 – The experimental power spectrum of a trapped bead (data) with the theoretical curve (line) from Equation (4.2) determined from a least squares fitting. The deviation of the theory and data at high frequencies is explained in section Chapter 5: .

The main advantages in the power spectrum method arise from the additional frequency content. Since the entire spectrum is fit with a model, deviations from ideality can be easily seen. Low frequency drift and other mechanical noise can be isolated by excluding their frequencies from the fit. Another advantage of this method is that it yields both the spring constant and the diffusion coefficient. Often the diffusion coefficient can be compared to a known or estimated value, providing additional data about the quality of the fit.

As with the equipartition method, this method only quantifies the harmonic region of the trap, yielding a local spring constant. Since the method is based on fitting the theoretical curve of Equation (4.2), it is crucial that the main feature of the function (i.e. the corner frequency) can be seen in the data. This brings up the issue of data acquisition rate, which can be a major limiting factor for this method. Since the Nyquist frequency determines the highest frequency in Figure 14, the corner frequency must be significantly lower than the Nyquist frequency to perform the calibration robustly.

Another drawback of this method is the additional computational steps required beyond the equipartition calibration. Computational time can be minimized by blocking the data and by keeping the number of data points to a power of 2. Also, aliasing and finite integration time of the detection change the shape of the theoretical model, causing error. These issues and possible solutions are discussed in detail in section Chapter 5: .

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This method is perhaps the most popular calibration method, because in practice it is nearly as simple as the equipartition method but is less sensitive to disruption from noise. For fast detection systems such as photodiodes, the effect of the integration time and data acquisition speed on calibration can often be neglected. For slower systems (i.e. video rates to 10 kHz) these issues can pose more severe problems, and may require a more complete theoretical model presented in section Chapter 5: to properly fit the data.

4.1.3. Drag Force Calibration The drag force method of calibration is a purely mechanical measurement able to quantify the entire trapping potential. It is based on the idea that moving a fluid with respect to a trapped bead will impart a force on the bead, which will displace it from the trap center. If the displacement can be measured and the force due to the fluid friction can be calculated, then a force vs. displacement curve can be built.

For micron size beads in water, the Reynolds number is many orders of magnitude below 1, which means that the system is overdamped and there is effectively no inertia. In this environment, the velocity of a particle with respect to a fluid is directly proportional to the force due to friction. The drag force is well described by Stokes’ formula for a sphere in an infinite fluid:

6 r γ = πη (4.4)

Where η is the fluid viscosity and r is the bead radius. If these two parameters are known or can be measured, then by moving the fluid or the bead at a known velocity, the force can easily be calculated.

In practice, this method first requires a way to move the bead or the fluid at a known velocity. Often it is easier to move the fluid filled stage than to translate the optical trap. Typically, this movement would be controlled with a piezo or other accurate motion control device. Movement of the stage with a constant velocity in one dimension will displace the bead from the trap center. It is important to have enough travel so that equilibrium can be verified and an average displacement can be recorded. This calibration is demonstrated in Figure 15 and Figure 17.

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Figure 15 – Raw data from a drag force calibration. The fluid velocity is controlled (bottom), causing a displacement of the bead from the trap center (top). With a known friction factor, this graph would produce one data point on a one­sided force vs.

displacement graph.

The main advantage to this method is that it can image the trap potential beyond the linear region (Figure 17). A smaller advantage is that it is less sensitive to error from mechanical noise than some other methods, especially if an average displacement can be recorded.

Several disadvantages exist for this method. First, it requires precise control of either the trap velocity or the fluid velocity, typically from a piezo device which can be very expensive. Second, it requires recording bead displacements at a variety of different velocities since each constant velocity only yields a single data point on a force vs. displacement graph. Considering this, performing the calibration can be quite time consuming unless some automated procedure is developed. Finally, there is an increased potential for error using this method. Error can come from several sources, including velocity, bead radius, viscosity, and displacement. Also, since the experiment is more involved, experimental errors or inconsistencies can cause problems.

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4.1.4. Recoil Calibration The recoil calibration is based on analysis of the recoil of a bead to the optical trap center after being released from outside the trap center. This mechanical measurement is based on a simple equation of motion, consisting of the force from the optical trap and a drag force (Figure 12). The inertial force is neglected due to the low Reynolds number as described previously. As discussed in section 4.1.3, the drag force on a sphere is proportional to the relative velocity between the sphere and fluid. In the general case of an arbitrary potential:

( ) OT OT

dU F x t

dx −

= = γ& (4.5)

If only the spring constant of the harmonic region is required, Equation (4.5) can be simplified by noting that FOT = ­kx(t), yielding a simple first order differential equation:

( ) dx k x t dt

= − γ

(4.6)

This can easily be solved for x(t):

( ) exp x t tk γ = −

(4.7)

which predicts the position of a recoiling bead as a function of time to be a decaying exponential.

The diffusion coefficient must be known or calculated to use this method. In the simple case, the exponential decay (in the harmonic region) can be fit with equation (4.7), quickly yielding the spring constant. In the general case, the potential relative to trap center can be calculated by integrating the velocities:

0

( ) (0) ( ') ' x

OT OT U x U x x dx = − γ ∫ & (4.8)

Using Equation (4.8), the trap potential can be completely reconstructed if the signal v(t) can be measured [Capitanio et. al, 2002].

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Figure 16 – An example of a recoil calibration within the linear regime. Here, a bead is released away from the trap center and the exponential decay to the trap center (inset – expanded scale of main figure) is fit to yield the time constant, which depends only on the

spring constant and friction factor.

Experimentally, a bead can be forced from the trap center by pulling it with a micropipette or by letting it deviate on its own from Brownian motion. When pulling a bead and releasing, it is important to ensure that there are no additional forces from the process of releasing. It is tempting to hold a bead by suction and subsequently release it by applying pressure, but this can cause errors due to the additional force of moving fluid nearby. Similarly, when the bead is in close proximity to another object, there can be hydrodynamic coupling.

Like the drag force method, this method has the ability to image the entire trap potential. However, this calibration can be difficult because it requires a simple way to hold and release a bead without introducing significant fluid flow. Also, for some types of video imaging it can be difficult to obtain significant data during the typically short recoil time.

4.1.5. Comparison of Calibration methods A summary of the four different calibration methods is given in

39

Table 1. It is important to note that the advantages and disadvantages can vary depending on the specific application and experimental apparatus. Also, several of the disadvantages of the noise based methods are eliminated when using the blur­corrected equipartition or power spectrum analysis (see section Chapter 5: ).

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Table 1 – Summary of calibration methods Calibration Method

Advantages Disadvantages

Equipartition § Fast § Simple

§ Harmonic regime only § Low frequency drift can cause error § Detection integration time can cause

error Power Spectrum

§ Fast § Includes diffusion

coefficient § Easier to isolate

mechanical noise

§ Harmonic regime only § Low frequency drift can cause error § Detection integration time can cause

error § Limited by Nyquist frequency

Drag Force § Not limited to harmonic regime

§ Less sensitive to error from mechanical noise

§ Requires precise motion control (expensive)

§ Time consuming § Many sources of error § Requires knowledge of diffusion

coefficient Recoil § Not limited to

harmonic regime § Simple for harmonic

regime

§ Requires method of releasing bead near trap

§ Time consuming § Accuracy may suffer without fast

detection § Requires knowledge of diffusion

coefficient

To verify the calibration methods, a comparison of three methods was done for a single bead at a single laser power. The trap stiffness was low enough that the effects due to aliasing and integration time could be neglected (see section Chapter 5: ). The equipartition, power spectrum, and force drag methods were used. The results of this comparison can be seen in Figure 17 below.

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Figure 17 – Comparison of three common optical trap calibrations, performed on a single bead at a single power. The three methods differ by about 5%.

All three methods yield similar results for the spring constant, differing by about 5%. It is not surprising that the equipartition method yields the lowest spring constant, since the variance can be overestimated in the presence of any mechanical noise (which is nearly impossible to eliminate completely). It is important to note that the drag data is not necessarily the actual force vs. displacement, since the creation of this curve from the raw velocity vs. displacement data required some advance knowledge of the friction factor for the bead. In this case, the friction factor was obtained from the fitting parameters of the power spectrum fit. However, it can also be calculated from the Stokes drag formula. Both estimates of the friction factor agree within about 10%.

4.2. Extent of the Harmonic Regime As demonstrated in Figure 17, the optical trap behaves like a linear spring for small displacements, but deviates from this ideality at larger displacements. This harmonic regime determines the effective range of this instrument, unless a more complicated calibration was to be used. From drag force data with 1 µm radius beads, it is clear that the linear approximation is valid until about 30% of the bead radius. For smaller particles (500 nm radius), another group measured a linear regime of about 50% of the bead radius

42

[Capitanio et. al, 2002]. For larger beads in the ray optics regime, the linear approximation extends to about 50% of the bead radius [Ashkin, 1992].

In the ideal case, experiments will be conducted entirely within the harmonic regime. However, in the force ramp mode there may be rare forces extending beyond this region, simply because of the stochastic nature of the interactions. If a calibration curve such as Figure 17 is known, then the force can be determined at any position.

4.3. Other Considerations

4.3.1. Hydrodynamics Bead hydrodynamics can play an important role in both trap calibration and experimentation. All of the calibrations described in this section assume that a free bead is being imaged. Ideally, free means that the bead is in an infinitely large fluid, but in reality it is only necessary to be a suitable distance from other surfaces. If a bead is close to a surface, the bead dynamics can change dramatically since the bead and the surface are hydrodynamically coupled. For a bead near a plane, the drag coefficient is described by [Happel and Brenner, 1991]:

3 4 5

6

9 1 45 1 1 16 8 256 16

r

r r r r h h h h

πη γ =

− + − −

(4.9)

Where h is the distance between the sphere and a planar surface. For a bead 3 microns in diameter, 30 microns from the surface, the drag coefficient is about 3% higher than predicted by Equation (4.4). In practice, it is a good idea to perform all calibration and experimentation away from the cover slip to minimize hydrodynamic effects.

4.3.2. Bead Size Optical trap stiffness has been shown to vary with bead size [Ghislain et. al., 1994]. For a laser with 1064 nm wavelength, the highest stiffness occurs for beads between 2 µm and 3 µm in diameter, which are commonly used in single molecule experiments, including the experiments in this thesis.

4.3.3. Height Dependence The properties of the optical trap have some height dependence, due to both hydrodynamic coupling and optical changes such as spherical aberrations [Rohrbach and

43

Stelzer, 2002]. Even when changes in hydrodynamic coupling are negligible, the trap stiffness generally changes with distance from the cover slip (Figure 18).

Figure 18 – Spring constant vs. bead height

Because of this height dependence, it is best to calibrate and perform the experiment at the same trap height, and to maintain a constant trap height throughout the experiment. Additionally, the experiments presented in this thesis were performed at a height near 30 µm, where changes in the bead height due to drift have little effect on the spring constant.

Also worthy of consideration is the change in the transverse spring constant in the axial direction. During experiments, it can be difficult to verify axial alignment of the two beads with nanometer precision. By eye, focus changes of 100 nm or less can be seen.

To measure the lateral spring constant for various axial positions, the 3D position of the bead is required. A concurrently developed instrument with 3D capabilities [Wong et. al., 2004; Heinrich et. al., manuscript under revision] was used to quantify this effect (Figure 19), showing a consistent lateral spring constant over a range of nearly 300 nm. This result ensures that our spring constant is conserved, even if the bead alignment is a little bit off.

44

Figure 19 – Measurement of the lateral spring constant using equipartition method at various bead heights using a 3D force probe [Heinrich et. al., under revision] with a

fixed optical trap.

45

Chapter 5: Optical Trap Calibration II: Calibrating With Image Blur

This chapter takes an in depth look into the effects of image blur on optical trap noise­ based calibrations. It has been reproduced from a publication submitted to Optics Express. Changes from the published form were made to fit the formatting of this thesis. Citations were moved to the end of the thesis, but appendices were kept within the chapter for completeness. Equations, figures and sections have been renumbered to fit the style of the thesis.

5.1. Abstract Dynamical instrument limitations, such as finite detection bandwidth, do not simply add statistical errors to fluctuation measurements, but can create significant systematic biases that affect the measurement of steady­state properties. Such effects must be considered when calibrating ultra­sensitive force probes by analyzing the observed Brownian fluctuations. In this article, we present a novel method for extracting the true spring constant and diffusion coefficient of a harmonically confined Brownian particle that extends the standard equipartition and power spectrum techniques to account for video­ image motion blur. These results are confirmed both numerically with a Brownian dynamics simulation, and experimentally with laser optical tweezers.

5.2. Introduction Investigations of micro­ to nano­scale phenomena at finite temperature (e.g. single­ molecule measurements, microrheology) require a detailed treatment of the Brownian fluctuations that mediate weak interactions and kinetics [Svoboda et. al., 1994; Mason et. al., 1995; Evans et. al., 1997; Collin et. al., 2005]. Experimental quantification of such fluctuations are affected by instrument limitations, which can introduce errors in surprising ways. Dynamical limitations, such as finite detection bandwidth, do not simply add statistical errors to fluctuation measurements, but can create significant systematic biases that affect the measurement of steady­state properties such as fluctuation amplitudes and probability densities (e.g. position histograms).

Motion blur, which results from time­averaging a signal over a finite integration time, can create significant problems when imaging fast moving objects. It is particularly relevant when measuring the position fluctuations of a Brownian particle, where even fast detection methods can have long integration times with respect to the relevant time scale, as we will demonstrate in this paper. Instrument bandwidth limitations that arise from motion blur affect a variety of fluctuation­based measurement techniques, including the quantification of forces with magnetic tweezers using lateral fluctuations [Strick et. al.,

46

1996], and microrheology measurements based on the video­tracking of small particles [Chen et. al., 2003]. The issue of video­image motion blur has recently been addressed in the single­molecule literature [Yasuda et. al., 1996] and in the field of microrheology, where the static and dynamic errors resulting from video­tracking have been carefully analyzed [Savin and Doyle, 2005; Savin and Doyle, 2005 (PRE)]. However, discussion has been notably absent in the area of ultra­sensitive force­probes, despite the significant effect that it can have on quantitative measurements. This paper focuses on the practical problem of calibration an optical trap by analyzing the confined Brownian motion of a trapped particle [Ghislain and Webb, 1993; Svoboda and Block, 1994; Gittes and Schmidt, 1998; Florin et. al., 1998; Berg­Sorensen and Flyvbjerg, 2004] in the presence of video­image motion blur.

In this article, we present a novel method for extracting the true spring constant and diffusion coefficient of a harmonically confined Brownian particle that extends the standard equipartition and power spectrum techniques to account for motion blur. In section 5.3 we describe how the measured variance of the position of a harmonically trapped Brownian particle depends on the integration time of the detection apparatus, the diffusion coefficient of the bead and the trap stiffness. Next, this theoretical relationship is compared with both simulated data (section 5.4) and experimental data using an optical trap (section 5.5). Practical strategies for trap calibration are given in the discussion section 5.6, where we show that motion blur is not a liability once it is understood, but rather provides valuable information about the dynamics of bead motion. In particular, we show how both the spring constant and the diffusion coefficient can be determined by measuring position fluctuations while varying either the shutter speed of the acquisition system or the confinement strength of the trap.

5.3. Bias in the variance of a harmonically trapped Brownian particle Detection systems, such as video cameras and photodiodes, do not measure the instantaneous position of a particle. Rather, the measured position m X is an average of the true position X taken over a finite time interval, which we call the integration time W . In the simplest model,

1 ( ) ( ) t

m t W X t X t dt

W − ′ ′ = ∫ (5.1)

where both the measured and true positions of the particle have been expressed as functions of time t . More complex situations can be treated by multiplying ( ) X t′ by an instrument­dependent function within the integral, i.e. by using a non­rectangular integration kernel.

47

We consider the case of a particle undergoing Brownian motion within a harmonic potential, 2 1

2 ( ) U x kx = . In equilibrium, we expect the probability density of the particle position to be established by the Boltzmann weight exp( ( ) ) B U x k T − / , where B k is the Boltzmann constant and T is the absolute temperature:

2 1 ( ) exp 2 2 X

B B

kx x k T k T k

ρ π

= −

/ (5.2)

The variance of the position should then satisfy the equipartition theorem,

2 2 var( ) B k T X X X k

≡ − = (5.3)

However, these equations do not hold for the measured position m X . In particular, motion blur introduces a systematic bias in the measured variance,

var( ) var( ) m X X ≤ (5.4)

Following standard techniques (e.g. [Oppenheim et. al., 1996]), the necessary correction can be calculated precisely as a function of the spring constant k , the friction factor of the particle γ , and the integration time of the imaging device W [Yasuda et. al., 1996; Savin and Doyle, 2005; Savin and Doyle, 2005 (PRE)]. First, we define the dimensionless parameter α by expressing the exposure time W in units of the trap relaxation time k τ γ = / , i.e.

W α τ

≡ (5.5)

Note that α can also be expressed in terms of the diffusion coefficient D by using the Einstein relation B k T D γ = / , i.e. ( ) B WDk k T α = / . Then as presented in appendix A (section 5.8), the measured variance is given by:

var( ) var( ) ( ) m X X S α = (5.6)

where ( ) S α is the motion blur correction function:

( ) 2

2 2 ( ) 1 exp( ) S α α α α

= − − − (5.7)

48

5.4. Numerical Studies To verify Eq. (5.6) numerically, we use a simple Brownian dynamics [Ermak and McCammon, 1978] simulation of a bead fluctuating in a harmonic potential. For each time step t ∆ , the change in the bead position x ∆ is given by a discretization of the overdamped Langevin equation:

det ( ) B

D x f x t k T

δ ∆ = + ∆ (5.8)

where ( ) x t δ ∆ is a Gaussian random variable with 0 x δ = and ( ) 2 2 x D t = ∆ δ , and

the deterministic force det f kx = − corresponds to a harmonic potential as in our calculation. Motion blur is simulated by time­averaging the simulated bead positions over a finite integration time W . To minimize errors due to discretization, the simulation sampling time is much smaller than both W and m Γ/ .

Figure 20a shows the simulation results for 3 different bead and spring constant settings as described in the caption. Agreement with the motion blur correction function ( ) S α of Eq. (5.7) is within the fractional standard error of the variance, 2 N / : [Kenney and Keeping, 1951]. We observe in Figure 20b that the distribution of measured positions is a Gaussian random variable with variance var( ) m B X k T k < / , and is therefore fully characterized by the mean (trap center) and the measured variance calculated in Eq. (5.6).

Figure 20 ­ (a) Brownian dynamics simulation results for measured variance as a function of exposure time. Data has been rescaled and plotted alongside ( ) S α , the motion blur correction function of Eq. (5.7), showing excellent agreement within the

expected error. The step size of the simulation is by 1 s µ , which is less than 0 01τ . for all

49

three simulations. The different simulation settings are: (i) 1 6 m µ . bead radius, 0 05 k = . pN/nm, 0 537ms τ = . , (ii) 0 4 m µ . bead radius, 0 05 k = . pN/nm, 0 134ms τ = . , (iii) (1 6 m µ . bead radius, 0 0125 k = . pN/nm, 2 148ms τ = . ) (b) Histogram of measured

positions for simulation run (c) for an exposure time of 4 ms. It is a Gaussian distribution as expected [Wang and Uhlenbeck, 1945]. The normal curve with the predicted variance is superimposed showing excellent agreement. The expected distribution for an ideal

“blur­free” measurement system is superimposed as a dotted line.

5.5. Experimental verification

5.5.1. Instrument description The optical trap is formed by focusing 1064 nm near­IR laser light (Coherent Compass 1064­4000M Nd:YVO 4 laser) through a high numerical aperture oil immersion objective (Zeiss Plan Neofluar 100x/1.3) into a closed, water filled chamber. Laser power is varied with a liquid­crystal power controller (Brockton Electro­Optics). This optical tweezers system is integrated into an inverted light microscope (Zeiss Axiovert S100).

The trapped bead is imaged with transmitted bright field illumination provided by a 100 W halogen lamp (Zeiss HAL 100). The image is observed with a high­speed cooled CCD camera with adjustable exposure time (Cooke high performance SensiCam) connected to a computer running custom data acquisition software [Heinrich, V.]. Each video frame is processed in real­time to determine the position of the trapped bead. Fast one­ dimensional position detection is accomplished by analyzing the intensity profile of a single line passing through the bead center. To increase the signal to noise ratio and the frame rate, the camera bins (i.e. spatially integrates) several lines about the bead center (32 in this experiment) to form the single line used in analysis. A third order polynomial is fit to the two minima corresponding to the one­dimensional “edges” of the bead, giving sub­pixel position detection with a measured accuracy of about 2 nm.

Tracking errors can be included in the measured variance by adding the parameter 2 ε to Eq. (5.6), i.e.

2 2

2

2 var( ) (1 exp( )) B m

k T X W k W W

τ τ τ ε

= − − − / +

(5.9)

2 ε can be interpreted as the measured variance of a stationary particle, provided there are no correlations between the tracking error and the measured positions. A detailed treatment of such particle tracking errors is provided in reference [Savin and Doyle, 2005], where the authors also stress the importance of using identical conditions of noise

50

and signal quality when comparing the 2 ε parameter between different experimental runs. Care was taken to achieve these conditions as is described below.

5.5.2. Experimental Conditions The sample chamber was prepared with pure water and polystyrene beads (Duke Scientific certified size standards 4203A, 3.063 µ m ± 0.027 µ m). Experiments were performed by holding a bead in the optical trap and varying the power and the exposure time. The bead was held 30 µ m from the closest surface, and the lamp intensity was varied with exposure time to ensure a similar intensity profile for each test. This ensured that the noise and signal quality between experimental runs was very similar, as validated by the results. For each test, both edges of the bead in one dimension were recorded and averaged to estimate the center position.

5.5.3. Experimental Results The one­dimensional variance of a single bead in an optical trap was measured at various laser powers and exposure times. Low frequency instrument drift was filtered out as described in Appendix B (section 5.9). For each power, measured variance vs. exposure time data was fit with Eq. (5.9) to yield values for the spring constant k , friction factor γ , and tracking error 2 ε . Error estimates in the variance were calculated from the standard error due to the finite sample size, and variations due to vertical drift.

Error in the fitting parameters indicate that the best estimates for γ and 2 ε occur at the lowest and highest powers, respectively. These estimates both agree within 2% of the error weighted average for all powers. For the nominal bead size, γ agrees with the Stokes’ formula calculation to within 11%, indicating a slightly smaller bead or lower water viscosity than expected. Additionally, the estimate of tracking error ε determined from the fit compares favorably with the standard deviation in position of a stuck bead, differing by about half a nanometer.

For a single bead observed under identical measurement conditions, γ and 2 ε are expected to remain essentially constant as the laser power is varied. Good consistency was found between the determined values of 2 ε from different experimental runs, due to the protocol of matching the signal strength between tests. While laser heating could cause γ to decrease with increasing power, this effect should be small for the < 500 mW powers used here [Peterman et. al., 2003; Celliers and Conia, 2000], so this effect was neglected.

Holding γ and 2 ε constant for all powers, the raw data was re­fit with Eq. (5.9) to yield k . The data for all 4 powers was error­corrected by subtracting 2 ε and was rescaled

51

according to Eq. (5.6) and Eq. (5.7). This non­dimensionalized data is plotted alongside the motion blur correction function in Figure 21, showing near­perfect quantitative agreement. This exceptional agreement further validates our treatment of γ and 2 ε . A plot of spring constant vs. dimensionless power is shown in Figure 22, demonstrating the discrepancy between the blur­corrected spring constant and naïve spring constant for different integration times. Even for a modest spring constant of 0.03 pN/nm and a reasonably fast exposure time of 1 ms, the expected error is roughly 50%. We also note that the blur­corrected spring constant increases linearly with laser power as expected from optical­trapping theory. Once confirmed for a given system, this linearity can be exploited to determine not only the spring constant as a function of power but also the diffusion coefficient of the bead. This is discussed in subsection 5.6.3, and presented in Figure 22.

Figure 21 ­ Fractional variance (var( ) var( )) m X X / vs. dimensionless exposure time W α τ = / for experimental optical trap data at 4 different powers. Overlaid on the data is

the motion blur correction function ( ) S α given by Eq. (5.7).

52

Figure 22 ­ Spring constant vs. power for a single bead in the optical trap. The naïve equipartition measured spring constant with 1 ms and 2 ms exposure times (red triangles and blue squares, respectively) is compared with the blur corrected spring constant (black circles). The dashed blue and red lines going through the uncorrected data

represent non­linear fits to the blur model assuming a linear relationship between k and laser power, i.e. k cP = , as discussed in subsection 5.6.3. The values obtained from these fits for c and γ agree within error with the “black circle” values obtained by

varying the exposure time.

When the data acquisition rate is sufficiently high relative to 1 k τ γ / = / , it is feasible to calibrate the trap using the bead position power spectrum, allowing comparisons to the previous results at low laser power. Power spectrum fitting with the blur­corrected and aliased expression (Eq. (5.23)) at the lowest power yielded both a spring constant and friction factor that agree with the blur­corrected equipartition values to within 1%. Fits of the same data using the naïve expression (Eq. (5.13)), not corrected for exposure time or aliasing) provided slightly worse results, overestimating the spring constant by 3% and the friction factor by 7%. (See Appendix C (section 5.10) for procedural details.)

For an additional check that does not rely on fluctuations, a purely mechanical test was performed and compared with the corrected power spectrum fit. This test consisted of a bead drop experiment to determine the bead radius and friction factor, and a trap recoil experiment to determine the spring constant. The bead drop was performed by releasing a bead and recording its average velocity over a known distance. The trap recoil experiment was performed by measuring the exponential decay of the same bead as it returned to the trap center after deviation in one dimension. This mechanical test agrees with the corrected power spectrum fit to within 5% for the determination of both the spring constant and the friction factor.

53

5.6. Discussion: Practical suggestions for calibrating an optical trap In this section, we present some practical techniques for measuring the spring constant k and diffusion coefficient D of a harmonically confined Brownian particle. We will assume that the temperature T is known. The approaches here are generic, and can be used even if the confining potential is not an optical trap (e.g. beads embedded in a gel, etc.) We will continue to treat the measured position as an unweighted time average of the true position over the integration time W (Eq. (5.1)), which is consistent with the experimental results for our detection system. In other situations, e.g. if the rise and fall time are not negligible relative to the exposure time, these equations and ideas can be readily generalized as noted in section 5.3.

5.6.1. Determining k from D and W If the diffusion coefficient D of the confined particle and the integration time W of the instrument is known, the correct spring constant k can be directly obtained from the measured variance var( ) m X by using Eq. (5.6). If the tracking error ε is significant, 2 ε should first be subtracted from the measured variance as in Eq. (5.9). While we cannot in general isolate k in this transcendental equation, it can easily be found numerically by utilizing a standard root­finding method. Alternatively, an approximate closed form solution for k is derived in Appendix D (section 5.11).

5.6.2. Determining k and D by varying W Even if the data acquisition rate of the system is not fast enough to permit a blur­ corrected power spectrum fit (as described in Appendix C), k and D can still be determined by measuring the variance at different shutter speeds and fitting to the blur­ corrected variance function. This technique is demonstrated in the experimental results section, and yields accurate measurements provided the integration time is not too much larger than the trap relaxation time (α is not much larger than 1). Practically speaking, this is a useful technique, as the maximum shutter speed of a camera is often much faster than the maximum data acquisition speed (e.g. it is much easier to obtain a video camera with a 0.1 ms shutter speed than a camera with a frame rate of 10 kHz). Furthermore, this approach for quantifying the power spectrum from the blur is quite general, and could be used in other systems. As long as the form of the power spectrum is known, the model parameters could be determined by measuring the total variance over a suitable spectrum of shutter speeds.

5.6.3. Determining k and D by varying k. Other approaches are possible if the confinement of the particles can be varied in a controlled way, i.e. by varying the laser power of the optical trap. If the spring constant varies linearly with laser power, (which is typically true and was confirmed for our system in subsection 5.5.3), the first observation is that the spring constant only needs to

54

be measured at a single laser power, as it can be extrapolated to other laser powers. Typically calibration should be done at a low power, as this usually increases the accuracy of both the power spectrum fit and the blur correction technique.

Linearity between the spring constant and laser power can be further exploited to determine both k and D by measuring the variance of a trapped bead at different laser powers but with the same shutter speed. Such data can be fit to the blur model (equation (5.6), recalling that ( ) B WDk k T α = / ) by introducing an additional fitting parameter c that relates the laser power P to the spring constant, i.e. we make the substitution k cP = , and perform a non­linear fit to var( ) m X vs. power data in order to determine c and D . Equivalently, we can express the naïve spring constant var( ) m B m k k T X = / as a function of c and P , and perform a fit to m k vs. P data as shown in Figure 22 of the experimental results subsection 5.5.3, where the viability of this method is demonstrated.

5.6.4. Design strategies for using the blur technique When using these motion blur techniques to characterize the dynamics of confined particles, we reiterate that it is the shutter speed and not the data acquisition speed that limits the dynamic range of a measurement. Thus, even inexpensive cameras with fast shutter speeds can make dynamical measurements without requiring the investment of a fast video camera. Alternative methods for controlling the exposure time are the use of optical shutters or strobe lights.

5.7. Conclusions We have experimentally verified a relationship between the measured variance of a harmonically confined particle and the integration time of the detection device. This yields a practical prescription for calibrating an optical trap that corrects and extends both the standard equipartition and power spectrum methods. By measuring the variance at different shutter speeds or different laser powers, the true spring constant can be determined by application of the motion blur correction function of Eq. (5.7). Additionally, this provides a new technique for determining the diffusion coefficient of a confined particle from time­averaged fluctuations.

The dramatic results from our experiment indicate that integration time of the detection device cannot be overlooked, especially with video detection. Furthermore, we have shown that motion blur need not be a detriment if it is well understood, as it provides useful information about the dynamics of the system being studied.

55

5.8. Appendix A: Calculation of the measured variance of a harmonically trapped Brownian particle

In this appendix, a derivation of the motion blur correction function Eq. (5.7) is presented. This quantifies how the measured variance depends upon the spring constant k , the diffusion coefficient of the particle D , and the integration time of the imaging device W (notation is as introduced in section 5.3). The derivation follows standard techniques (e.g. [Oppenheim et. al., 1996]) and is similar to calculations presented in references [Wang and Uhlenbeck, 1945; Yasuda et. al., 1996; Savin and Doyle, 2005; Savin and Doyle, 2005 (PRE)]. For completeness, we present the calculation in two different ways: a frequency­space calculation that convolves the true particle trajectory with the appropriate moving average filter, and a real­space calculation using Green’s functions. An expression for the modified power spectrum of the harmonically confined bead that accounts for the effects of filtering and aliasing is included in the frequency­ space calculation.

5.8.1. Frequency­space calculation The measured trajectory of a particle in the presence of motion blur ( ) m X t can be calculated by convolving the true trajectory ( ) X t with a rectangular function,

( ) ( ) ( ) ( ) ( ) m X t X t H t X t H t t dt ′ ′ ′ = ∗ ≡ − ∫ (5.10)

where ( ) H t is defined by:

1 0 ( )

0 elsewhere

t W H t W

< ≤ =

(5.11)

The integral is taken over the full range of values (i.e. t′ is integrated from −∞ to +∞ ), which is our convention whenever limits are not explicitly written. The width of the rectangle W is simply the integration time as previously defined. This convolution acts as an ideal moving average filter in time, and is consistent with the integral expression for

( ) m X t given in Eq. (5.1).

Taking the power spectrum of Eq. (5.10) yields:

2 2 2 ( ) ( ) ( ) ( ) m m P X H X ω ω ω ω ≡ = % % % (5.12)

56

Where the Fourier transform is denoted by a tilde, e.g. ( ) ( )exp( ) X X t i t dt ω ω = ∫ % , and ω is the frequency in radians/second. The theoretical power spectrum ( ) P ω is given by:

2

2 2 2

2 ( ) ( ) B k T P X k

γ ω ω γ ω

≡ = +

% (5.13)

where γ is the friction factor of the particle, and is related to the diffusion coefficient by the Einstein relation B k T D γ = / . This power spectrum has been well­described previously [Svoboda and Block, 1994; Gittes and Schmidt, 1998; Wang and Uhlenbeck, 1945], and is derived subsequently in the real­space calculation.

The power spectrum of the moving average filter can be expressed as a squared sinc function:

2 2 sin( 2) ( )

2 W H W ω

ω ω

/ = / % (5.14)

Using Parseval’s Theorem and integrating the power spectrum ( ) P ω yields the true variance of ( ) X t ,

1 var( ) ( ) 2

B k T X P d k

ω ω π

= = ∫ (5.15)

which is in agreement with the equipartition theorem. Similarly, we calculate the measured variance var( ) m X as a function of the exposure time W and the friction factor γ by integrating the power spectrum of the measured position (Eq. (5.12)):

1 var( ) ( ) 2 m m X P d ω ω π

= ∫ (5.16)

2

2

2 (1 exp( )) B k T W k W W

τ τ τ

= − − − /

(5.17)

where k τ γ = / , the trap relaxation time. Writing this formula in terms of the dimensionless exposure time,

W α τ

≡ (5.18)

57

and the variance of the true bead position var( ) B X k T k = / yields:

var( ) var( ) ( ) m X X S α = (5.19)

where ( ) S α is the motion blur correction function:

( ) 2

2 2 ( ) 1 exp( ) S α α α α

= − − − (5.20)

5.8.2. Blur­corrected power spectrum Often, trap calibration is performed by fitting the power spectrum of a confined particle. Here we provide a modification to the standard functional form ( ) P ω that accounts for both exposure time effects and aliasing. Combining expressions (5.12), (5.13) and (5.14), we can see the effect of exposure time on the measured power spectrum:

2

2 2 2

2 sin( 2) ( ) 2

B m

k T W P k W

γ ω ω

γ ω ω / = + /

(5.21)

Additionally, the effect of aliasing can be accounted for:

aliased ( ) m s n

P P n ω ω +∞

=−∞

= + ∑ (5.22)

2

2 2 2

sin(( ) 2) 2 ( ) ( ) 2

s B

n s s

n W k T n k n W

ω ω γ γ ω ω ω ω

+∞

=−∞

+ / = + + + /

∑ (5.23)

where s ω is the angular sampling frequency (i.e. the data acquisition rate times 2π ). Aliasing changes the shape of the power spectrum, so neglecting it when fitting can cause errors. The sum in Eq. (5.23) can be calculated numerically and fit to experimental data. It is typically sufficient to calculate only the first few terms.

It is important to note that aliasing does not affect our result for the measured variance, Eq. (5.17). Aliasing shifts power into the wrong frequencies, but does not change the integral of the power. Hence, var( ) m X is unchanged. A detailed discussion of power spectrum calibration with an emphasis on photodiode detection systems is given in reference [Berg­Sorensen and Flyvbjerg, 2004].

58

5.8.3. Real­space calculation Since a Brownian particle follows a random trajectory ( ) X t , the measured position m X is a random function of the true position of the particle at the start of the integration time, i.e.

0 0 0

1 ( ) ( | ) W

m X x X t x dt W

= ∫ (5.24)

where 0 ( | ) X t x is the actual position of the bead at time t given that it is at position 0 x at time zero, and W is the integration time as defined previously. In other words, even with knowledge of the initial particle position, it is not possible to predict what the measured position will be. However, the distribution of m X is well­defined, and one can determine its moments.

The variance of the measured position is given by:

2 2 var( ) ( ) ( ) m m m X X X X X ≡ − (5.25)

Notice that to calculate the ensemble average … , we must average over both the random initial position X , and the measured position for a given initial position ( ) m X x . For the harmonic potential 2 1

2 ( ) U x kx = , ( ) 0 m X X = by symmetry, so the variance reduces to

2 0 0 0 var( ) ( ) ( ) m X m X x X x dx ρ = ∫ (5.26)

where 0 ( ) X x ρ is the probability density of the initial position, and the integral is taken over all space (consistent with our previously stated convention). In equilibrium, 0 ( ) X x ρ is simply the Boltzmann distribution given in Eq. (5.2).

Using Eq. (5.24), we express 2 0 ( ) m X x as the double integral

2 0 1 0 2 0 1 2 2 0 0

1 ( ) ( | ) ( | ) W W

m X x X t x X t x dt dt W

= ∫ ∫ (5.27)

2

2 1 1 0 2 0 1 2 2 0 0

2 ( | ) ( | ) W t

t t X t x X t x dt dt

W > = ∫ ∫ (5.28)

59

In the second step, the ensemble average is brought into the integral, and the averaging condition 2 1 t t > is added, which changes the limits of integration.

The time­ordered auto­correlation function 2 1

1 2 ( ) ( ) t t

X t X t >

can be calculated using the

Green’s function of the diffusion equation for a harmonic potential, ( ) 0 0 | x t x t , , ρ . The Green’s function represents the probability density for finding the particle at position x at time t given that it is at 0 x at time 0 t . It can be found by solving the diffusion equation

2

2 B B

D D D kx k t x k T x k T ρ ρ ρ ρ

∂ ∂ ∂ = + +

∂ ∂ ∂ (5.29)

with the initial conditions 0 0 ( ) ( ) x t x x ρ δ , = − . The solution to this problem is well­ known [Wang and Uhlenbeck, 1945; Doi and Edwards, 1986] and is given by:

( ) ( ) 2 0 0 0 0

0 0

exp( ( ) ) 1 | exp 2 ( ) 2 ( ) B B

k x x t t x t x t

k TV t t k TV t t k

− − − / , , = − − − /

τ ρ

π (5.30)

where we have defined the dimensionless function:

( ) 1 exp( 2 ) V t t τ = − − / (5.31)

As before ( ) B k T kD k τ γ = / = / . Notice that this is simply a spreading Gaussian distribution with the mean given by the deterministic (non­Brownian) position of a particle connected to a spring in an overdamped environment, and with a variance that looks like free diffusion at short time scales (i.e. initially increasing as 0 2 ( ) D t t − ), but exponentially approaching the equilibrium value of B k T k / on longer time scales. The time­ordered auto­correlation function can be written as follows:

2 1 1 2 1 2 1 1 0 2 2 1 1 1 2 ( ) ( ) ( | 0) ( | )

t t X t X t x x x t x x t x t dx dx

> = , , , , ∫∫ ρ ρ (5.32)

Putting in the Green’s function of Eq. (5.30) and evaluating the integrals gives the result:

2 1

2 1 1 2 0 2 1 2 1

( ) ( ) ( ) exp( ( ) ) exp( ( ) ) B t t

k TV t X t X t x t t t t k

τ τ >

= − + / + − − / (5.33)

where ( ) V t and τ are as defined above.

60

Carrying out the double time integral in Eq. (5.27), followed by the integral over the initial position 0 x of Eq. (5.26) we obtain the final result for the measured variance:

2

2

2 var( ) (1 exp( )) B m

k T X W k W W

τ τ τ

= − − − /

(5.34)

This reproduces the result of the frequency­space calculation presented in Eq. (5.17). The ideal power spectrum can be obtained from the position auto­correlation function of Eq. (5.33). We determine the long­time limit of the auto­correlation function by letting 1 t τ ? , which yields the simplified equation:

1 2 2 1 ( ) ( ) exp( ) B k T X t X t t t k

τ = − − / (5.35)

Next, by taking the Fourier transform of this equation with respect to 2 1 ( ) t t − we obtain the standard result of Eq. (5.13).

5.9. Appendix B: High­pass filtering in variance measurements Calculation of the variance requires special attention, since low frequency noise or drift can inflate the variance dramatically, causing an underestimation of the spring constant. A high pass filter can be used to remove low frequency noise, but the use of any ideal filter lowers the variance by neglecting the contribution from the removed frequencies (note Eq. (5.15)).

To reliably estimate the variance while accounting for low frequency drift, we first progressively high­pass filter the data over a range of increasing cut­off frequencies. A plot of measured variance vs. cutoff frequency (Figure 23) clearly shows a linear trend at frequencies below the corner frequency (fc=k/2πγ). However, as the filtering frequencies approach zero, drift causes the measured variance to increase beyond its expected value. By applying a linear fit and extrapolating to the 0 Hz cutoff, we can reliably estimate the “drift­free” variance of bead position.

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Figure 23 ­ Experimentally measured variance as a function of the high pass filter cutoff frequency shows a linear relation (line), which can be extrapolated to 0 Hz to reliably estimate the drift­free variance. The variance without filtering (cross) is 2 100 nm , while

the extrapolated variance (star) is 2 78 5 nm .

5.10. Appendix C: Experimental power spectrum calibration Power spectrum calibrations were performed by fitting the one­sided power spectrum with Eq. (5.23). The original 65536 data points taken at ~1500 samples per second were blocked into 128 non­overlapping segments. The power spectrum of the blocks were calculated separately and averaged to produce the data in Figure 24. This procedure is well described in the literature [Gittes and Schmidt, 1998; Berg­Sorensen and Flyvbjerg, 2004]. This data was fit with the blur­corrected and aliased model of Eq. (5.23) and compared with the commonly used non­corrected power spectrum of Eq. (5.13), both with and without aliasing. The quality of the fit to Eq. (5.23) was further investigated by examining the fractional deviation in the power (the measured data divided by the model fit) as in reference [Berg­Sorensen and Flyvbjerg, 2004]. A scatter plot and histogram of the fractional deviation is presented in Figure 25. The histogram agrees well with a Gaussian distribution with a standard deviation of 1/√128 (see reference [Berg­Sorensen and Flyvbjerg, 2004] for a thorough discussion of power­spectrum fitting, including the expected scatter from unity).

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Figure 24 ­ A log­log plot of the one­sided power spectrum (dots) for a trapped bead, with theoretical models produced from a least squares fit to the data (blur­corrected and aliased, Eq. (5.23) solid line; naïve, Eq. (5.13) blue dashed line; naïve aliased, green dotted line). The effect of the motion blur correction function ( ) S α is readily apparent

from the clear discrepancy between the solid red and dotted green lines.

Figure 25 ­ Fractional deviation of the power spectrum data obtained by dividing the experimentally measured values (dots in Figure 24) by the fit obtained with the blur­

corrected and aliased model (solid red line in Figure 24). Left: Scatter plot demonstrating the quality of the fit; the two dashed red lines indicate the estimated

standard deviation from unity of 1 128 / [Berg­Sorensen and Flyvbjerg, 2004]. Right:

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Histogram of the fractional deviation data overlaid with a Gaussian distribution with a standard deviation of 1 128 / (solid red line).

Accounting for tracking error in the power spectrum fit is more difficult than in the equipartition case, requiring knowledge of the frequency dependence of the error. To investigate this in the current study, the power spectrum of a stationary bead was subtracted from the calibration power spectrum. We found that the fit parameters remained practically unchanged (within 2%), allowing us to neglect tracking error in our power spectrum fits at low power. It should be noted that in other situations (e.g. different bead size or power), modifications to the power spectrum due to tracking error could be significant.

5.11. Appendix D: Approximate analytical expression for k When W is not significantly larger than the trap relaxation time, i.e. W Wk α τ γ = / = / is not much larger than 1, an approximate version of equation (5.6) can be inverted to give a closed form solution for k . First, we use a Padé approximation to express the motion blur correction function as:

2 1 2 15 60 ( ) 1 5

S α α α α

− / + / ≈

+ / (5.36)

Substituting this expression into equation (5.6) yields a quadratic equation that is easily solved for k . This results in the following approximation for the true spring constant:

1 2 2 2 2

30

2 15var( ) 225var( ) 240 var( ) 11 B

m m m

k T k DW X X DW X D W

/

≈ + + + −

(5.37)

The Padé approximation is good to within 3% for 3 α < , which corresponds to a blur correction factor of (3) 0 46 S . B . In other words, if the uncorrected equipartition method gives a spring constant which is within a factor of 2 of the true value, this approximation formula should be accurate to within 3%, as we have tested numerically.

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Chapter 6: Forced Unfolding of the Spectrin Repeat

6.1. Abstract Single molecule measurements of forced protein unfolding can yield insight into the energy landscapes which govern the transition between the folded and unfolded state. In this study, the kinetic behavior of spectrin repeat unfolding is quantified using an optical trap based force probe. The spectrin repeat is a small (~106 amino acids) molecule bundled into three antiparallel alpha­helices.

An engineered poly­spectrin molecule was pulled from end to end at force loading rates spanning several orders of magnitude (1­500 pN/s). Unfolding was revealed by an abrupt change in length between two beads spanning the molecule, corresponding to an increased contour length. The force extension behavior of the molecule was measured and modeled with a worm like chain, yielding a full contour length of 36nm per domain and a persistence length of 0.76 nm. Histograms of unfolding forces were collected for each rate and fit with a simple model to yield kinetic parameters that govern the interaction, namely a force scale of 2.5 pN and a spontaneous off­rate of 0.003 s ­1 .

6.2. Introduction The spectrin repeat, a bundle of three antiparallel alpha­helices [Speicher and Marchesi, 1984; Yan et. al., 1993; Pascual et. al., 1997], forms the structural basis for a variety of cytoskeletal proteins (e.g. spectrin, alpha­actinin). These proteins are constructed from multiple repeats strung together in series, and are often found in structures exposed to mechanical stress (i.e. cell cortex, muscle sarcomere, etc)[Pascual et. al., 1997; Djinovic­ Carugo et. al., 2002], suggesting that the spectrin repeat is well adapted to supporting biological loads.

In erythrocytes, the spectrin protein (a ~200 nm long tetramer containing a series arrangement of multiple spectrin repeats in each of the four subunits) forms a two­ dimensional hexagonal network beneath the plasma membrane, which is responsible for the unique mechanical properties of the red cell [Alberts et. al., 1994; Mohandas and Evans, 1994]. Disruption of the spectrin network has been associated with a variety of disease states which exhibit abnormal mechanical properties (e.g. sickle cell anemia, spherocytosis, etc) [Mohandas and Chasis, 1993].

While the macroscopic properties of the red cell have been widely studied for decades, the molecular mechanics of spectrin have more recently been examined by single molecule techniques. The advent and progression of ultra­sensitive force probes (i.e.

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optical trap, AFM, BFP [Evans et. al., 1995], magnetic pullers, etc) has recently allowed exploration of weak intermolecular (i.e. receptor­ligand [Merkel et. al., 1999], DNA­ protein [Yin et. al., 1995]) and intramolecular (i.e. protein unfolding/refolding [Rief et. al., 1997; Oberhauser et. al., 2001; Fernandez et. al., 2004], DNA unzipping [Bockelmann et. al., 2002], and RNA hairpin unfolding/refolding [Liphardt et. al., 2001]) interactions.

Previous experiments pulling on single spectrin molecules have already shown that native spectrin unfolds under stress in well defined units at a force significantly lower than other proteins with β sheet structures [Rief et. al., 1999; Law et. al. 2003]. The relative instability of folded spectrin repeats implies that spectrin unfolding may be a physiologically relevant mechanism for supporting and regulating stress in the red cell membrane [Lee and Discher, 2001]. Additionally, studies with spectrin repeats in series have suggested cooperativity in the unfolding of adjacent domains [Law et. al., 2003]. Ensemble solution measurements of the unfolding and refolding kinetics have been performed [Scott et. al., 2004a&b, 2005; Batey et. al., 2006], but they do not reveal how transition rates depend on force, or other mechanical details such as compliance.

In this study we investigated the kinetics of forced unfolding of R16 α­spectrin domains with an optical trap based force probe. Single­molecule spectrin constructs were immobilized between two beads and pulled apart at loading rates spanning several orders of magnitude to elucidate the underlying kinetic behavior. In the Methods section (6.3) the instrument and experimental procedures are described. In the Results section (6.4), the measurements are presented and analyzed to determine mechanical and kinetic properties of the spectrin domain. These results are compared with previous studies and their implications discussed in the Discussion section (6.5).

6.3. Materials and Methods

6.3.1. Instrument Description The force probe layout and principle of operation are shown in Figure 26. The optical trap holds the probe bead and is formed by focusing near­IR laser light (Coherent compass 1064­4000M) through an oil immersion objective (Zeiss Plan Neofluar 100x/1.3) into a water filled chamber. The chamber is open on one side to accommodate the micropipette, which holds the test bead and is mounted to a one­dimensional piezo stage (Physik Instrumente P­753.12C) and a 3­dimensional micrometer stage (Newport 461 series). The entire system is integrated into an inverted light microscope (Zeiss Axiovert S100).

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Figure 26 illustrates the general principle of operation of the force probe. The schematic spring is drawn for visualization purposes only, to show the lateral (x­axis) direction of force application by the 3D optical trap utilized in this setup. The functionalized probe

bead reports the formation and failure of single biomolecular bonds as well as intramolecular changes such as protein unfolding and refolding. The test bead is mounted to a piezo translator, which moves the test bead into contact with the probe

bead, and retracts upon real­time detection of the touch. The formation of a bond, as well as the force it experiences, is reported by the displacement of the probe bead from the trap center. Intramolecular transitions are revealed by abrupt changes in distance

between the two beads.

Transmitted light images are recorded using a fast CCD camera (Cooke high performance Sensicam). Real­time one­dimensional position detection of a both beads is accomplished by analyzing the intensity profile of a single line passing through the axis of symmetry. To increase the signal to noise ratio, the camera bins several lines about this axis (32 in this experiment) to form the single line used in analysis. A third order polynomial is fit to the two minima corresponding to the one­dimensional “edges” of the beads, giving sub­ pixel position detection with a measured accuracy of about ±2 nm, at a rate of ~1500 frames/second.

6.3.2. Trap Calibration Force calibration of the optical trap system was performed using the blur­corrected power spectrum fit [Wong and Halvorsen, 2006]. The spring constant was determined at low power and linearly extrapolated for higher powers to get the desired spring constant. The spring constant was typically 0.05 pN/nm, but was varied between 0.04­0.07 pN/nm depending on the loading rate (i.e. higher loading rates had higher spring constant). The

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linear region of the trap was determined to be at least 600 nm for the beads used in this experiment.

6.3.3. Sample Preparation Engineered spectrin constructs were kindly provided by Jane Clark (MRC Centre for Protein Engineering, University of Cambridge Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, UK). The construct consists of alternating spectrin R16 domains (from chicken brain α­spectrin) and I27 domains with 4 of each domain type. The ends of the spectrin titin [(R16)­(I27)]4 construct have been engineered to facilitate the linking of the molecule to a force probe. At one end of the construct there is an accessible Cysteine residue, suitable for functionalization via its sulf­hydryl group. At the opposite end of the molecule there is a his­tag, a sequence of 6 histidines that binds to the anti­his antibody. Details about the protein expression and purification methods used for similar spectrin molecules is given in reference [Scott et. al., 2004].

The purified protein was covalently linked to streptavidin with a maleimide reaction on the cysteine (Sigma streptavidin­maleimide S9415­2MG). This streptavidin­spectrin­ 6xHIS construct was immobilized on biotin beads and tested against anti­histidine beads. The two sets of beads were prepared as follows:

Borosilicate glass beads (Duke Scientific, ~2.5 µm diameter) were first cleaned in a mixture of ammonium hydroxide, hydrogen peroxide, and water at boiling temperature. After several washes with nanopure water, the beads were coated with mercapto­silane groups (United Chemical Technologies mercapto­propyl­trimethoxy silane) and baked to finish the covalent bonding of silane.

These silanized beads were used to prepare biotinylated beads by reacting with maleimide­PEG3400­biotin (Nektar Therapeutics). To prepare beads with covalently linked anti­histidine, the silanized beads were reacted with maleimide­PEG3400­penta­ anti­histidine, previously formed by reacting maleimide­PEG3400­NHS (Nektar Therapeutics) with penta­anti­histidine (Qiagen).

6.3.4. Experimental Conditions Force ramp experiments were performed by bringing the test bead into feedback­ controlled contact with the probe bead and subsequently retracting the test bead at a constant speed. Experiments were performed at room temperature (25ºC) in Hepes buffer (10 mM Hepes and 69 mM NaCl) with 0.1% BSA. A force calibration was performed for each bead prior to the experiment. Both the calibrations and experiments were performed at roughly 30 µm from the cover glass to avoid surface interactions and hydrodynamic coupling between the bead and surface.

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Alignment of the two beads was established and maintained throughout the experiment by two feedback controlled piezo devices (Physik Instrumente P­170.07 and P­721.10). Approach, touch, and retract conditions were established and controlled by custom software. For these experiments, approach rate was 2000 nm/s, touch force was 7­10 pN, and touch time was 0.1 s. Retraction rate was varied between 1 pN/s and 500 pN/s, while keeping touch and approach conditions constant.

6.4. Results Single spectrin molecules were pulled from end to end in a force ramp at 4 different rates. The position of the two handle beads were tracked, allowing measurement of both the force applied to the spectrin molecule and the change in distance due to conformational changes such as unfolding. For each loading rate, hundreds of tests were performed to collect sufficient statistics to characterize the stochastic process of forced unfolding.

Each test had 3 possible outcomes: no attachment, attachment without unfolding, and attachment with unfolding. Figure 27 shows a data trace with unfolding of 4 spectrin domains. Tests with attachment but no unfolding were deemed either non­specific interactions (i.e. not spectrin), or spectrin that detached from one bead before unfolding.

For each test with attachment, there was opportunity for the unfolding of up to four spectrin domains. Domain unfolding was marked by an abrupt increase in the distance of the two handle beads (Figure 27). While unfolding of less than four domains was common, unfolding of more than 4 domains was rare, and could be attributed to imperfections in the chemical linkages or the purified protein.

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Figure 27 – A data trace shows unfolding of four spectrin domains. In the force graph (top), the events are marked by a small (~1 pN) drop in force. In the tether length graph (bottom), the events are easily distinguishable as the length abruptly increases due to the

unfolding.

6.4.1. Spectrin domain compliance The change in length (∆L) from each unfolding event was measured directly. A histogram of these lengths for a narrow range of unfolding forces can be fit with a Gaussian (Figure 28). The peak identifies the best estimate of ∆L at that force, while the width reflects the measurement error of the instrument in the limit where each histogram represents a single unfolding force. Here, the standard deviation of 2 nm compares favorably with the instrument accuracy claimed in section 3.4.2.

To analyze the compliance of a single spectrin domain, data was sorted by unfolding force and divided into sections containing 50 data points each. Histograms of ∆L for each force were constructed and fit with a Gaussian as in Figure 28. A force extension curve (Figure 29) was constructed by plotting average unfolding forces against the total

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domain length (i.e. ∆L + Lo), where the length of a folded spectrin domain was taken as 4.2 nm [Altmann et. al., 2002]. A worm­like chain model [Bustamante et. al, 1994]:

( ) 2 4 1 ( ) 1

4 1 / B k T x F x b L x L

= + − −

(6.1)

was fit to the data, yielding a contour length of 36.2 nm and a persistence length of 0.76 nm.

Figure 28 – A histogram of the change in tether length due to a spectrin domain unfolding at a force between 12.5 and 15

pN.

Figure 29 – Force vs. displacement for an unfolded spectrin domain (dots) with a least squares fit using a worm­like chain

model (line).

Since the compliance of spectrin is not a stochastic parameter, the expected length at a given force can be used to effectively filter the data. This ensures that the force data is representative of the full domain unfolding, and not some other anomalous event such as peeling or disentanglement.

6.4.2. Spectrin domain unfolding kinetics Unfolding force statistics were collected and organized by loading rate. All events for a given loading rate were treated equally since each domain experiences roughly the same force history throughout a single test. This effectively treats one pull of four domains the same as four pulls of one domain.

The raw data was filtered to decrease the amount of data representing either non­spectrin interactions or multiple spectrin interactions. The worm­like chain model determined in Figure 29 was used to filter out events which did not have the expected force­extension

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relationship. The expected length at a given unfolding force was compared with the actual unfolding length, and data falling outside a tolerance (+/­ 5 nm) was filtered out. Additionally, tests with more than 4 unfolding events or only a single event were not included. More than 4 events indicates multiple molecules, while single events give no confidence that spectrin was unfolded as opposed to a single anomalous event such as unpeeling. The final histograms are shown in Figure 30.

Figure 30 – Histograms of spectrin unfolding forces for 4 different loading rates: 500 pN/s (1 st from top), 100 pN/s (2 nd from top), 10 pN/s (3 rd from top), and 1 pN/s (4 th from

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top) with the universal distribution (8.10) overlaid on the data with fb=2.5 pN and koff 0 =0.003 s ­1 .

The universal distribution for forced dissociation (Equation (8.10)) was used to simultaneously fit the histograms (Figure 30). The agreement between the data and the model indicates that the interactions governing spectrin unfolding can be treated as a single sharp energy barrier. Based on the best fit, the spontaneous off­rate was 0.003 s ­1 and the force scale was 2.5 pN.

While the model encapsulates most of the data and fits the peaks of the distributions well, there is still data which is outside the bounds of the model. Several explanations exist for these small deviations between the model and the data. Statistical errors (~sqrt(N)) can cause the bin heights to vary based on the number of data in each bin. While this clearly does not account for most of the data outside of the model, small outlying bins may be explained in this way. Also, uncertainty in force due to tracking and calibration uncertainty can cause the distributions to be wider than expected, but the small tracking error (section 3.4.2) and accurate calibration method (section 5.5.3) limit this error to no more than 1­2 pN. Data which still lies outside the model can be explained by rare multiple­spectrin bonds or other anomalous events. Multiple spectrin molecules in parallel between the beads would cause some sharing of the force, leading to higher unfolding forces. Forced unfolding at higher forces than predicted by the model may be explained in this way. Anomalous unfolding events such as a spectrin domain peeling from the bead surface could cause a force measurement which does not reflect forced unfolding of spectrin. By applying more aggressive filtering criteria to the data in Figure 30 (i.e. tighter length tolerances or eliminating the first events), significant outlying data is removed while still preserving the peaks in Figure 31. This suggests that some anomalous events have artificially broadened the width of the distributions.

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Figure 31 – The most likely unfolding force vs. logarithm of the loading rate (dots) shows a linear relationship (line) with a force scale of 2.5 pN and a stress­free off rate of 0.003

s ­1 .

6.5. Discussion This study shows that R16 unfolding is characterized by a small force scale (2.5 pN) and an unstressed off rate of 0.003 s ­1 . The unfolded domain exhibits worm­like chain behavior, with a contour length of 36.2 nm and a persistence length of 0.76 nm. The low unfolding force and the small force scale emphasize the high force sensitivity and large span of loading rates required to make these kinetic measurements.

The compliant behavior of the α­spectrin repeat was consistent with several other studies. The change in contour length measured in this study agrees with the expected change in contour length of 32 nm predicted by structural data and measured with AFM [Rief et. al., 1999; Altmann et. al., 2002]. This suggests that the unfolded state observed is the fully denatured state. In the AFM study by Rief et al., the force­extension curve is matched to a WLC fit with a persistence length of 0.8 nm, which is extremely close to the 0.76 nm obtained here. Another AFM study found the average actual end to end length change from a single unfolding event of a β­spectrin domain to be 22 nm [Law et. al., 2003] and found a persistence length of 0.5 nm to fit the data well. Considering the low unfolding force (~25 pN) found in the study, the 22 nm change in length is consistent with the worm­like chain model found in this study, which predicts a length change of ~23 nm at a force of 25 pN.

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Comparison of kinetic data with previous studies is complicated by the variety of spectrin domains studied and the dramatically larger stiffness of the AFM (typically 2­4 orders of magnitude stiffer than optical trap, i.e. 10­100 pN/nm vs. 0.01­0.1 pN/nm). Measurement of unfolding kinetics by stopped­flow fluorimetry and CD have shown variations over several orders of magnitude in the solution unfolding rate between chicken brain α­ spectrin domains R15, R16 and R17 [Scott et. al., 2004]. The large stiffness of the AFM (typically ~10 pN/nm) results in faster force loading rates than those accessed by optical trap. Additionally, the soft polymer in series with the stiff cantilever can cause nonlinear force loading when a constant pulling speed is applied.

The unfolding forces measured here are significantly lower than AFM measurements of 4 consecutive α R16 domains [Lenne et. al., 2000]. Lenne et. al. found widely distributed unfolding forces (SD = ~15 pN) with a mean above 60 pN even for the slowest pulling rate of 300 nm/s. Unfortunately, neither the cantilever stiffness nor the force­loading rate was given. However, a 10 pN/nm cantilever (typical for these types of experiments) would suggest a force loading rate of 3000 pN/s, roughly an order of magnitude faster than the fastest rate of 500 pN/s in this study. Based on the force scale and off rate determined in this study, an unfolding force of less than 35 pN would be expected for a loading rate of 3000 pN/s unless there is a rate dependent change in bond dynamics between these rates. Such dynamical transitions have been observed in other systems [Merkel et. al., 1999; Evans et. al., 2004]. Apparent differences in the unfolding force could also come about from interactions with adjacent spectrin domains. Scott et. al. showed that R16 is less kinetically stable when isolated than when in series with R17 [Batey et. al., 2005]. Similarly, it is possible that 4 R16 domains in series is more stable than 4 isolated R16 domains, though this has yet to be studied directly.

Other studies unfolding different spectrin domains observed unfolding at forces similar to those presented here [Rief et. al., 1999; Law et. al., 2003]. Rief et. al. pulled on a α­ spectrin R13­R18 construct and found unfolding forces between 25 and 35 pN for nominal loading rates on the order of 1000 pN/s and 10,000 pN/s (determined by multiplying pulling speed by cantilever stiffness). At the slower rate, they found a peak unfolding force of ~27 pN, which is comparable to our peak at 500 pN/s. At 10,000 pN/s, the model presented in this study predicts an unfolding force of ~36 pN, and Rief et. al. measured ~32 pN. This small discrepancy could be a result of the polymer loading characteristics, causing the true force loading rate to be smaller than the nominal loading rate.

The unstressed off rate of 0.003 s ­1 found in this study is in close agreement with previous solution studies of R16 [Scott et. al., 2004]. Scott et. al. used stopped­flow fluorescence and CD to estimate a solution off rate of 0.0026 s ­1 . This close agreement suggests that the energy barrier imaged by this study is the same energy barrier that dominates solution unfolding. One AFM study has estimated a solution off­rate of 3E­5, though the authors admit that the errors are too large to make an accurate measurement of this quantity [Rief et. al., 1999].

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In contrast to other AFM measurements of spectrin, one study found evidence to support the existence of an intermediate unfolded state with a contour length of about 16 nm using AFM and MD simulations [Lenne et. al., 2000; Altmann et. al., 2002]. The study presented here, however, found little evidence to support this claim. While some data showed an unfolding length lower than the main peak, it was a very small percentage of the total data. This is in great contrast with the study presented by Lenne et. al., where roughly half of the data is short unfolding and half is long unfolding.

It is still possible that a short­lived intermediate state exists, with the transition from the intermediate state to the fully unfolded state occurring on a time scale shorter than ~1ms in the optical trap experiment. The drop in force that results from a sudden increase in tether length that accompanies an unfolding event depends upon the stiffness of the force probe. In the case of the AFM experiments, forced unfolding of spectrin adds 20­30 nm of length, causing complete or near complete relaxation to zero force because of the stiff spring. This is in marked contrast to the optical trap, where the same forced unfolding can lower the force one piconewton or less due to the soft spring (Figure 27). As an example, if unfolding to an intermediate state occurs at 20 pN, the intermediate state would be subject to a ~19 pN force in the case of the optical trap, and a ~0 pN force in the case of the AFM. Since the off­rate is exponentially dependent on force (Equation (2.4)), this may dramatically affect the lifetime, causing the state to be observable in the AFM experiment and not in the optical trap experiment. Further investigation would be required to examine this possibility. As an alternative explanation, it is possible that the intermediate state can only occur when domains are directly in series, as opposed to the isolated R16 domains used in this study.

6.6. Conclusions This single molecule study of the spectrin repeat has provided a quantitative look at the kinetic behavior of full domain unfolding, establishing both a force dependence (fβ = 2.5 pN) and the off rate (koff o = 0.003 s ­1 ). These parameters hold predictive value for a range of force loading rates between 1 pN/s and 500 pN/s, and likely extend to arbitrarily slower rates due to the close agreement between the off rate determined in this study and measurements made in solution. Further investigations are required to explore faster force loading rates as well as exotic behavior such as possible intermediate states.

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Chapter 7: Future Work: Improving Instrument Operation and Throughput

7.1. Overview This chapter outlines the remaining work required to increase the throughput of the instrument presented in Chapter 3: to one capable of testing at least 10 compounds per day. The envisioned instrument would be highly automated, requiring minimal setup (e.g. inserting bead samples and setting the testing parameters) and yielding the off­rate and force dependence of the interactions. Several developments in this thesis such as automated alignment and semi­automated data analysis represent a first step toward this end, but significant work remains.

Dramatically increasing the throughput of the device poses a unique set of engineering challenges. Using the current instrument in force ramp mode as a framework, the entire experimental process must be automated, from setup and calibration to data analysis. The experiment must also be streamlined to minimize dead time, as roughly 100,000 tests per day are needed to obtain dynamic force spectra of 10 compounds. The challenges involved in this work as well as some possible solutions are discussed below.

7.2. Experiment Preparation

7.2.1. New Chamber Design The simple chamber presented in section 3.9.1 would prove insufficient for an automated device. A new chamber design would be required to keep the beads and the test area physically separated, and allow easy exchange of beads and possibly the surrounding media. One approach is the design of a microfluidic chamber, which would have microfluidic channels to flow single beads into the test area on demand (Figure 32). To achieve single bead precision, it may be necessary to experiment with channel dimensions. Additionally, recent developments in microfluidics such as valves [Unger et. al., 2000] could prove useful, allowing the full or partial restriction of channels to better control the flow of beads. This idea could easily be scaled up to accommodate more wells and microfluidic channels.

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Figure 32 – A sketch of a microfluidic test chamber (below) with two holding wells and a test area connected by microfluidic channels. The wells would be loaded with

functionalized beads and could be deposited into the test area by applying pressure to the wells. The interface plate (above) facilitates the connection of tubing for applying

pressure to flow beads or change the media.

Another approach is a PDMS chamber with a raised channels connecting the test area to individual holding wells (Figure 33). Beads would be transported via optical trap, which could raise a bead through the raised channel and into the test area.

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Figure 33 – A sketch of a test chamber with 4 holding wells connected to a test area by raised channels. The raised channels prevent beads from leaving the holding wells, but

allow transport via optical trap.

A third possibility is to eliminate the external pipette completely and instead integrate it into a closed chamber (Figure 34). With this chamber design, the chamber itself would move back and forth rather than moving the pipette with respect to the stationary chamber. This pipette integration could be accomplished either by using a glass capillary micropipette (section 3.9.2) and fixing it to the chamber, or by constructing a pipette­like device as part of the test chamber. The main challenge of this design is developing a manufacturing technique to embed a pipette or pipette­like device.

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Figure 34 – A sketch of a closed test chamber with integrated pipette. Four holding wells are connected to a test area by raised channels as in Figure 33. An additional hole

provides an interface to the pipette for pressurization.

Each of these chambers could be built out of PDMS using soft lithography techniques [Xia and Whitesides, 1998] and fixed atop standard glass cover slips. The top surface could either be another glass cover slip, or a machined interface plate as shown in Figure 32.

7.2.2. Automated Bead Loading To begin an experiment, one bead needs to be aspirated into the pipette and the other needs to be in the optical trap. This process is typically done manually, but it too could be automated. Bead recognition software would have to be developed to move the stage and pipette to the proper location to obtain the beads.

7.2.3. Improved Chemistry One of the greatest challenges in single molecule experiments is establishing reliable chemical linkage of molecules to surfaces such as glass, metal, or plastic. Several methods exist, some of which are described in section 3.8.

True high­throughput testing would require uniformity among beads as well as repeatability between batches of beads. Recently, functionalized beads have become a commercial product offered by several companies (e.g. Bangs Laboratories, Spherotech, Microspheres­nanospheres). With proper quality control, these mass produced beads could be a useful starting point for protein linkage. Some of these companies also offer protein linking kits, which simplify the process.

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Additionally, the development and use of long polymer linker molecules could increase the quality of data from single molecule experiments by providing a molecular signature. For example, if two interacting proteins are each linked to beads via a 100 nm long linker, failure to see a 200 nm length before rupture would indicate an interaction other than the protein­protein interaction of interest.

7.3. Instrument Use To successfully test 10 compounds per day, the instrument must be able to perform roughly 100,000 tests without user intervention. As with the data presented in this thesis, experiments must be performed at several loading rates to determine both the effect of force and an estimate of the off rate. Assuming that 3 different loading rates per compound is sufficient (e.g. 10, 100, and 1000 pN/s), 30 histograms need to be populated. With a 10% attachment frequency, 100,000 tests would be more than adequate, yielding over 300 data per histogram.

7.3.1. Increasing the Maximum Force To ensure that a wide variety of compounds can be tested with loading rates between 1 pN/s and 1000 pN/s, the maximum force of the instrument needs to be increased to 100­ 200 pN, from a current maximum of ~40 pN.

Optical trap force can be increased in a variety of ways, the most obvious being the use of higher laser powers. Currently the maximum usable laser power is limited by infrared absorption in the objective, which will burn at significantly higher powers. Unfortunately, the exact maximum intensity for most objectives is not known (or available from the manufacturer), since it typically requires the destruction of an expensive piece of equipment. Considering this, it is likely that our objective could withstand significantly higher powers than typically used. Indeed, I have successfully operated the laser at 33% higher power than used in the experiments in Chapter 6: , and even higher powers for short periods of time. I suspect that a 50% increase in maximum power will not destroy the objective.

Another way to increase the force is to simply exploit the nonlinear region of the optical trap. The current instrument is limited to the linear region of the trap, which can be as little as 1/3 of the bead radius, while the maximum force occurs farther away from the trap center, at approximately the full bead radius (Figure 17). Using the full extent of the trap potential requires a mechanical calibration instead of the currently used thermal noise calibration. A computer controllable piezo stage would allow a simple calibration procedure to yield a force vs. displacement curve extending beyond the linear region. The resulting curve could be used to determine the necessary nonlinear position vs. time protocol required to generate a linear force ramp for DFS studies.

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Using these two methods in tandem, the maximum force could easily be increased to 100 pN, and possibly as high as 200 pN. Higher forces would likely require custom high power laser objectives.

7.3.2. Automated Calibration Currently, calibration is performed by using the blur­corrected power spectrum fit from section 5.10. Custom software has already been developed to find the spring constant from the position of a trapped bead. However, the user is responsible for obtaining a bead, moving it away from the cover glass, starting data acquisition, running the fitting program, and adjusting the laser power to obtain the desired spring constant. This process would be relatively simple to automate with software control, requiring only a desired spring constant. The piezo controlled focus would bring the bead to the calibration height, the data acquisition would begin, the calibration software would measure the spring constant, and the laser power would be adjusted to obtain the desired spring constant.

7.3.3. Automated Alignment Automated alignment software has already been developed to ensure proper bead alignment from test to test (section 3.7.2). This software would require minor improvements for a high­throughput instrument such as improved feedback speed, and feedback based on if beads are in contact or not.

7.3.4. Shortening Experimental Time When considering 100,000 tests per day, streamlining the time per test is critical. Each test consists of an approach until contact, a pause, and a retraction at the desired ramp speed. The approach and contact time are typically independent of the desired force­ loading rate, and take roughly 1 second per cycle. The retraction time obviously depends on the ramp speed, and can thus vary between 0.1 and 30 seconds per cycle. With the current setup, I estimate that 100,000 tests divided between 3 loading rates would take nearly 4 days if operating continuously. There are several things that can be done to decrease the time per cycle, fitting 100,000 tests into under 16 hours.

First, the beads should start in close proximity, limiting the travel distance on the approach. Currently, beads start roughly 3 microns apart to accommodate the auto­ alignment software. With improvements in this software, the beads could start 1 micron apart. With a reasonable approach speed of 4 um/s, this decreases the approach time to 0.25 seconds. Adding a typical pause time of 0.1 seconds, this brings the ramp­ independent time to 0.35 seconds.

Second, the software should be modified to detect when the beads are in contact and start a new cycle when they are not. Currently, the ramp portion of the cycle operates the

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same way for every test, regardless of if an attachment formed or not. For slow ramps this can increase the experimental time enormously, since attachments are generally rare. For example, assume a 1 pN/s ramp is applied to a 30 pN maximum force. If this ramp is run 10 times, resulting in one attachment that breaks at 15 pN, the ramp time would take 5 minutes. If the ramp could be automatically aborted when the beads are not in contact, this time could be decreased to just over 15 seconds, or 20 times faster!

7.3.5. Forced Rupture of Strong Attachments One problem which is currently solved manually is the detachment of beads that become stuck to each other. Sometimes an unusually strong attachment (e.g. multiple bond) between beads prevents detachment with the optical trap. Typically this is solved by the very crude method of physically tapping the pipette assembly or by using the piezo to apply a small translation very quickly. The sudden motion detaches the beads, and the probe bead is “sucked” back to the optical trap center.

To solve this problem in an automated way requires a large impulse of force once the probe bead nears the extent of the optical trap (typically ~ 1 bead radius). The large impulse of force could come either by a sudden but small piezo movement or by a quick pulse of high power laser light. If the probe bead remained attached beyond a threshold position, the force impulse would be applied, causing the beads to separate.

7.4. Data Analysis

7.4.1. Automated Raw Data Analysis Raw data of force vs. time is currently processed using custom semi­automated software developed in labview (section 3.9.5). While this greatly accelerates data analysis, it still requires the user to identify the time and force of prominent features such as bond rupture with cursors. However, further development could eliminate the user input to provide fully automated data analysis. For simple adhesion experiments, the rupture force can be easily identified since the force drops precipitously afterwards. The simplest method would be to identify the maximum force in each trace, but this includes the noise in force due to tracking error. The averaged force noise could be subtracted to provide a better estimate, or more sophisticated algorithms that find the maximum of a best fit line could be developed. This automated raw data analysis would yield a distribution of rupture forces for further analysis.

7.4.2. Determination of Kinetic Parameters To determine the off rate and force dependence of the interaction, the distributions of rupture forces need to be fit with a model (e.g. (8.10)). For the simplest case described in

83

section 2.2, fitting the model to the data could be automated. The peaks of the distributions could be plotted vs. the logarithm of the loading rate as in Figure 31 to determine the off rate and force dependence. For more complex interactions, the rupture force distributions would have to be interpreted on a case by case basis.

7.5. Other Considerations

7.5.1. Parallelizing Performing multiple experiments in parallel would significantly increase the throughput of the device. This could be accomplished by creating multiple traps with an acousto­ optic modulator [Svoboda and Block, 1994] or more sophisticated holographic techniques [Grier, 2003]. The main drawback is that the total power remains the same, with the traps splitting the power. To be useful, this would require getting an order of magnitude more laser power through the objective, which is not possible in the current configuration. Optics with the capability of handling 10 Watts or more of laser power would be needed. Additionally, significant development work would be required to expand current software to parallel processes and to create an array of devices to hold the test beads in precise alignment.

7.5.2. Packaging The prototype instrument described in Chapter 3: was constructed on an optical table with breadboard style optical components to facilitate quick changes and experimentation with different configurations. Further engineering design could easily reduce the size of the instrument by integrating components into a single package. Additionally, properly enclosing the laser could make it in to a class 1 instrument, eliminating the need to wear laser goggles.

7.6. Summary Using the instrument presented in Chapter 3: as a template, I have outlined the necessary steps to dramatically increase the throughput to accommodate at least 10 compounds per day. Much of the remaining work consists of automating processes through software design, developing more reliable and simpler chemistry procedures, and designing a test chamber to facilitate automated testing.

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Chapter 8: Appendices

8.1. Appendix A: Chemical Kinetics Consider a single molecular complex A­B formed from individual components A and B. Let S be the probability of being in the bound state (A­B), noting then that 1­S must be the probability of being in the unbound state (A & B). The off rate is represented by koff and the on rate by kon. The rate of transitions between states is governed by the following equation, which notes that the rate of change of survival probability is dictated by the rates at which things bound become unbound and vice versa.

( ) ( ) ( )[1 ( )] off on dS k t S t k t S t dt

= − + − (8.1)

External forces on the A­B complex will alter the on and off rates, and in the case where the force changes with time, koff and kon are functions of time. In typical DFS experiments, complex A­B is pulled apart with force and there is little chance of rebinding once the complex dissociates. This simplifies equation (8.1) to:

( ) ( ) off dS k t S t dt

= − (8.2)

Where koff can be approximated using Kramers’ theory [Kramers, 1940; Hanggi, 1990]:

( ) ( ) exp off off f t k t k f β

≈

o (8.3)

Where fβ is the force scale for the interaction given by B

ts

k T x

. Integrating equation (8.2)

and noting the boundary condition that S(0)=1 (i.e. starting with complex A­B at time=0):

1 0

' ( ') ' '

S t

off dS k t dt S

= − ∫ ∫ (8.4)

The solution of this equation depends on the specific form of koff. In the case of constant force, koff is independent of time. For a linear force ramp, koff is a function of time determined by:

85

0

( ) exp ( ') ' t

off S t k t dt

= −

∫ (8.5)

For the case where the force is constant in time, the expression reduces to a simple decaying exponential with a time constant given by 1/koff. For a linear force ramp, force and time are related by:

( ) f f t r t = (8.6)

Where rf is the force­loading rate. Using this, expression ((8.5)) now becomes:

'

0

( ) exp ' f r t t f

off S t k e dt β

= −

∫ o (8.7)

Integrating, we get:

( ) exp 1 exp off f

f

k f r t S t

r f β

β

= −

o

(8.8)

Using equation (8.2), we can determine the probability density function:

( ) exp 1 exp f off f off

f

r t k f r t dS p t k dt f r f

β

β β

= − = + −

o

(8.9)

Similarly, this probability density function can be written as a function of force:

( ) exp 1 exp off off

f f

k k f dS f f p f df r f r f

β

β β

= − = + −

o

(8.10)

Which describes the expected force histogram for a DFS experiment. The most likely breaking force can be determined by noting that it occurs at the peak in the probability density:

* ln f

off

r f f

f k β β

=

o (8.11)

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8.2. Appendix B: Instrument Setup Guide

1) Clean microscope objective and replace immersion oil. 2) Lower the objective to its lowest position. 3) Turn on microscope lamp, computers, and cameras. 4) Start joystick program, feedback program, and experiment program. 5) Place the prepared chamber on the stage with the open end facing the 3D

translator. 6) Adjust the rough focusing knob until the oil contacts the bottom of the chamber. 7) Adjust the fine focusing knob until the bottom of the chamber is in focus. 8) Prepare Kohler illumination. 9) Attach the prepared micropipette to the chuck assembly. 10) Secure the chuck assembly to the x­piezo translator. 11) Secure the entire assembly to the 3D translator. 12) Use the 3D stage micrometers to adjust the micropipette position until it is in

focus near the center of the field of view. 13) Lift the micropipette from the surface with the micrometer. 14) Don goggles and turn on the laser, power controller, and shutter (make sure it is

closed). DO NOT USE EYEPIECES WHILE LASER IN USE. 15) Adjust the laser power to the desired value using the laser power supply and the

power controller. 16) Use the joystick to move the stage into one of the bead areas. 17) Identify a suitable bead. 18) Lower the micropipette and partially aspirate the bead by applying a negative

pressure in close proximity. 19) Lift the micropipette from the surface with the micrometer. 20) Use the joystick to move the stage to the other bead area. 21) Find and center a suitable bead, focus up a micron or more and open the shutter to

trap the bead. 22) Raise the focus at least 10­15 microns and move the stage to the bead­free area in

the center of the chamber (moving fast will prevent beads near the surface from becoming trapped).

23) Adjust the tilt micrometer on the beam expander until the bead is centered along the 16th vertical line from the top (if using 32 line binning, otherwise divide bin width by 2 to find center).

24) Adjust the masks on the Sensicam to perform FFR. The top one should be just above the top line of pixels, and the bottom one is binning lines plus transit lines lower. For instance, in the typical case of 32 binning and 11 transit lines, the edge of the lower mask should be 43 lines from the top.

25) Activate the Fast Frame mode by pushing the FFR button.

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26) Perform bead calibration with the focus 20 µm or more from the cover slip (see Chapter 5: Optical Trap Calibration).

27) Calculate spring constant and enter value into force ramp or force clamp program. 28) Align the two beads along the axis of motion using both coarse micrometer

adjustments and fine piezo adjustments. 29) Activate the feedback program (see section 4.7 for details) 30) Enter touch and retract parameters into the program. 31) Press “go” to start the experiment.

88

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