From Physics 111-Lab Wiki

All pages in this lab. Note To print Full Lab Write-up click on each link below and print separately

I. Optical Trapping (OTZ)

II. Staff Sign-Off Sheet (OTZ) and Pre-Lab Questions

III. Laser safety form

Reprints and other information can be found on the Physics 111 Library Site

Before The Lab

Note: This lab should be available soon. See Don Orlando if you are interested.

Before using the apparatus in this experiment, you must complete training in the safe use of lasers detailed on the Laser Safety Training page. This includes readings, watching a video, taking a quiz, and filling out a form.

Discuss the Physics about this experiment with the faculty or the GSI's in the 111-Lab before starting.

You should keep a laboratory notebook. The notebook should contain a detailed record of everything that was done and how/why it was done, as well as all of the data and analysis, also with plenty of how/why entries. This will aid you when you write your report.

Discuss the pre-lab questions with an instructor, and have the Staff Sign-Off Sheet (OTZ) signed.

Introduction

What is an Optical Trap?

An optical trap or "tweezers" is a device used to apply piconewton sized forces and make precise measurements on a scale of roughly one micron. It can be created by applying a precisely focused laser onto a dielectric material. It allows scientists to make very detailed manipulations and measurements on several objects in the field of cell biology and thus acts as a major tool in biophysics. They are used in biological experiments ranging from cell sorting to the unzipping of DNA and also in physical applications such as atom cooling.

History

Arthur Ashkin discovered the method of optical trapping in 1970. He calculated that the momentum from a high power laser, focused entirely onto a micron bead would propel the bead forward with 100,000 g's of acceleration. Taken by curiosity he performed this experiment and found that not only was the intended bead pushed downstream by the laser but also that other beads in his solution were highly attracted to the beam-path and flew in laterally from other parts of his slide. He then created the first working trap by using two opposing laser beams. At one point a bacterium that had contaminated a sample flew into the trap and was trapped, thus instigating the trap's revolutionary use in cell biology. Today optical traps are used extensively in both atom-trapping experiments and in Biophysics labs worldwide. Local UCB uses include Steven Chu's Nobel Prize winning work as well as the Jan Liphardt and Bustamente labs.

The Physics Behind Trapping

A ray diagram showing the light-bending forces active in the stability point of the Optical Trap

In order to trap a particle we need to create a stable equilibrium. The force that will be counteracting the movement in this case is the change in momentum (remember that force is defined to be dp/dt) of the laser light path (light carries momentum) as the trapped particle refracts and bends the light in various ways. In order to understand how it is stable, we only need to consider a couple of test cases.

In the figure to the right we see an illustration of two possible laser-particle set-ups. the red region represents the laser at its focus point, the blue ball is the particle, and the black arrows are representative light rays whose thicknesses correspond to their intensities (note that the beam is brightest at its center).

In case (a) we see that the particle has been moved slightly to the left, the two black arrows refract through the particle and bend inwards. The reactionary force vectors on the particle are also included in the image. We see that because the particle is slightly to the left of center, the right ray is more intense (and thus carries more momentum) than the left ray. As a result of the ray's bending to the left, the particle will be pushed to the right in order to conserve momentum. Thus, a perturbation to the left causes a right directed force back towards equilibrium.

In case (b) the particle is dead center and will not be pushed left or right. It is still reflecting some of the laser light hitting it dead on however and may be pushed "downstream." To counter act this force we note that the two beams, though now symmetric, are still being bent inwards. This slight bending creates a force that pulls the bead back towards the laser source. This only happens very near the focus of the laser and only when the light comes in at an extremely crossed angle or at a high numerical aperture.

If a picture is worth a thousand words then a java applet must be worth a million. Inspect this applet to get a feel of how the light-bending forces work. Be sure to adjust the numerical aperture at the bottom in order to obtain a working trap.

Prerequisite Reading Materials

1. Arthur Ashkin, Acceleration and Trapping of Particles by Radiation Pressure, Physical Review Letters Vol 24 Number 4 Jan 1970. This is the original paper debuting the practice of optical trapping using two opposing lasers.
2. K. C. Neuman and S. M. Block, Optical Trapping, Review Article, Review of Scientific Instruments 2004. An excellent resource on optical traps though at a fairly technical level.
3. MIT's Optical Trapping Experiment, a sister lab to our own.
4. J. Bechhoefer, S. Wilson, Faster, cheaper, safer optical tweezers for the undergraduate laboratory. Am. J. Phys., 70 (4), Apr. 2002. Read Optical tweezer theory and Applications, part B and Appendix for Trapped-Particle Statistical Analysis explanation.
5. Arthur Ashkin, Optical trapping and manipulation of neutral particles using lasers, Proc. Natl. Acad. Sci. Vol 94 1997. An excellent overview of tweezers including their development and applications throughout history. A fun read.
6. Optical Tweezers from Wikipedia.
7. F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw Hill, New York, 1965. A useful reference, available in the physics library, for understanding the autocorrelation function and Power Spectrum Density: Chap. 15, especially Secs. 6 and 10.
8. J. W. Shaevitz A Practical Guide to Optical Trapping, a less technical description of the optical set up and detection. Not everything here is relevant to us but it serves as an excellent starter guide.
9. Jerome Williams, Samuel A Elder. Fluid Physics for Oceanographers and Physicists, 1989 Pergamon Press Inc., New York. A useful reference for the physics of incompressible flow involved in the Stokes Drag Method.
10. Fungal Cell Biology Group at Edinburgh's site that contains videos of an optical trap in active use in mycology experiments.
11. F. Gittes, C. F. Schmidt, "Interference model for back-focal-plane displacement detection in optical tweezers." Optics Letters, 23(1):7-9, 1998.

Apparatus

Safety Measures

The laser used for trapping in this experiment has a power output of up to 200 mW and a wavelength of 845nm, which is near Infrared. Even a brief exposure to the focused beam at this power can cause permanent damage to the retina of your eye. Because the beam is invisible, you could be exposed without even realizing it. For this reason, the beam path is shielded wherever it is tightly focused. The safety training and measures below are essential for your protection.

1. Do not operate the laser unless you have completed the laser safety training and submitted the signed laser safety form and quiz to the course staff.
2. Never place your hands or any reflective material, such as rings or watches, in the path of the laser. Due to the invisibility of the infrared laser, its position cannot be seen and hence, it may be scattered in undesired, potentially harmful directions. Accidental exposure to the laser may cause blindness.
3. Do not open the lid of the black acrylic safety shield with the laser on. The box and the shield on the front of the microscope enclose the beam path of the laser. An electric interlock should automatically cut power to the laser if the lid is opened, but it is best not to count on this to protect you.
4. Never bypass any safety device.

LASER

The Light Amplification by Stimulated Emission of Radiation diode has a maximum power output of 200 mW, though after collimation only a fraction of this power is retained. The diode produces 845nm wavelength light, which is near Infrared and is chosen so as not to heat up trapped living organisms. This Diode Laser is coupled to an optical fiber. If this fiber is broken or kinked it could cause backwards reflections of the beam, destroying the laser and, thus, the experiment. Do not touch the laser or any metal components it is in contact with because any tiny amount of static discharge, much below what you can feel as a shock, can destroy the laser. Be careful when handling anything in the lasers optical path as adjustment will cause misalignment. If one optical device is misaligned it is much easier to correct than if multiple are misaligned. So if the laser becomes misaligned and/or is not trapping beads well, do not try to fix it, but notify the staff and realignment procedure will be arranged, which will likely delay experimentation for several days. It is important that you AVOID TAMPERING WITH THE OPTICAL PATH in any way.

• How does a diode laser work as compared to a typical optically pumped laser you may have seen elsewhere in the lab?

LASER operation

There are two controllers needed for proper laser operation. The top controller in the stack of 2 controllers is the Temperature Controller, also known as the TEC. This has a preset current limit, I_LIM, of 1.4 A and the resistance, T_ACT, should be around 10.0 kΩ, and always within the range 9.5 to 10.5 kΩ. This should already be set, but it is important to make sure this is not changed. To change the resistance, the TEC must first be 'enabled' go to the T_SET mode in the display section and turn the knob to just below 10 kΩ. The TEC, when set properly and 'enabled', will control the temperature of the laser diode when in operation. It should not need to be adjusted.

The adjustment for the laser diode power is via the Laser Diode Controller (LDC). Once the TEC has been set and 'enabled' the LDC can be used. Before 'enabling' the LDC, turn the gray rotational knob counterclockwise until the stop point and make sure the black box enclosing the laser path is closed. The LDC should be run in the current (I) mode and has a current limit I_LIM set to 285 mA which is 95% of the maximum current the laser can handle. Under the display menu, scroll down to the I_LD. This tells the actual current being delivered to the laser and is what you should watch when adjusting the laser during operation, once 'enabled'. After the TEC is set and enabled, the laser can finally be 'enabled' at which point the I_LD should jump to 1.6 mA or so. Making sure the black box is closed and objective in place, the laser knob can be turned on. If the max current, I_LIM, is reached, a 'beep' will sound and the current will not increase anymore. This should occur around the 285 mA setting. The laser should be operated slightly below the limit. At the highest point where the red light is NOT on is the highest safe operating current (around 285 mA).

As can be seen by the diagram of Current Vs. Power, the laser has a linear relationship between current delivered and power output to the objective. This is the approximate power that will enter the objective and govern how the trap behaves and should be used for any necessary power calculations. The linear relation can not be extrapolated well below 100 mA or above 300 mA.

• Briefly, what are the functions of the TEC and LDC?

Quadrant Photo-Diode (QPD)

Schematic of the Quadrant Photo Diode position detection system. The signal from the diodes are amplified by low-noise preamplifies and then networked to calculate the X and Y position of the incident light beam. Fig. due to Allersma et al.

The infrared laser scatters off of a particle and is reflected by a dichroic mirror onto the Quadrant Photo Diode (QPD). The QPD is 4 photodiodes in a quadrant formation to allow X and Y position calculation. Within a certain range of light intensities, the output voltage of a photodiode scales linearly with the intensity of light incident upon the diode. The light incident upon each quadrant in the QPD generates a voltage. The analog circuitry then outputs a voltage Vx and Vy which are proportional to the actual X and Y position of the incident beam only around the center of the trap. As the light scatters in a predictable way off of the spherical beads, this information can be used to recover actual bead position within a narrow range around the center of the trap. Further information can be found in reference: Gittes and Schmidt.

• Explain briefly how a QPD works in the lab write up, without going into details about the circuitry.

Optics

1. LASER OPTICS The laser exits the diode and travels through a fiber optic cable to the back of a Beam Expander. This increases the width of the beam so that when it eventually enters the objective it fills the lens thus creating a more narrow waist when it exits the objective, thus creating a stronger, more localized trap. The beam, though pictured as red, is near infrared, and thus invisible. Using a special type of photosensitive dye the beam can be detected. The light is reflected off of 4 mirrors, the third of which is a Dichroic Mirror. Then the beam passes through the high Numerical Aperture Objective creating a narrow beam waist where particles are trapped. After this waist, however, the LASER acts like a point source and is highly divergent. The laser then passes through a condenser, which turns the highly divergent beams into parallel rays. This signal is then bounced off of another Dichroic Mirror and onto the surface of the QPD, producing a voltage signal. When a particle interferes with the laser by getting trapped at the waist, it causes the light to scatter, affecting the position of the laser on the Quadrant Photo Diode surface.

• How does the Dichroic mirror work?

2. VISIBLE LIGHT OPTICS As the light is infrared and fairly low powered, it cannot illuminate the microscope to allow us to see what is happening. Rather, a blue LED light is used to illuminate the contents of the slide. The LED light is point source and so it is focused with a lens. This light then passes through the a Dichroic Mirror and then the condenser, further focusing the LED light onto the objects in the microscope slide. The light then passes through the objective, magnifying the contents of the slide by 100. The light then bounces off of a mirror below the stage and into the box, through another Dichroic Mirror, splitting off from the laser light path. The blue light then passes through a 200mm 'eyepiece' which further magnifies the picture. After another reflection, this light hits the CCD camera lens and creates the picture.

• Why is blue as opposed to white light used to illuminate the beads?

User Controls

The User will be able to adjust the X, Y and Z position of the stage. Familiarize yourself with the knobs for the X, Y and Z position on the trap using the picture at right. The Z direction will be the primary tool to focus the camera view and laser beam. By changing the Z direction of the stage, the waist of the laser beam can be moved up and down through the fluid on the slide allowing beads to be trapped away from either of the glass barriers. The X and Y position will allow stage movement which will be used for searching the microscope slide for particles and QPD calibration. The picture above shows the knobs for adjusting the various parameters. Note that the Z position can only be adjusted by hand by the lower of the two vertical knobs. The upper knob, labeled DO NOT TOUCH is for adjusting the light incident upon the QPD. If this knob is tampered with it will change the focus of the laser on the QPD and a staff member will need to readjust the focus. The X and Y position can be adjusted very roughly by hand by turning the wheels connected to the red Piezo Electric Picomotors. The window, Stage Control, a part of the People’s Optical Trapping Program Suite, can be used to adjust the X and Y positions by moving them at a specified frequency of steps for a certain total number of steps. These can also be controlled using the joystick.

• Why is a piezo electric crystal used to generate small movements of the stage?

Orientation Procedure

How to pipette

Forward Technique:

1. Depress the push button to the first stop.
2. Dip the tip under the surface of the liquid in the reservoir to a depth of about 1 cm and slowly release the push button. Withdraw the tip from the liquid touching it against the edge of the reservoir to remove excess liquid.
3. Deliver the liquid by gently depressing the push button to the first stop. After a delay of about one second, continue to depress the push button all the way to the second stop. This action will empty the tip.
4. Release the push button to the ready position.

If necessary, change the tip and continue pipetting.

• Adjust the volume of fluid dispensed by rotating the thumbwheel until the desired volume

Never allow liquid to enter the body of the micropipette

• To avoid contaminating the nanoparticle suspensions, wear gloves and put a fresh filter tip on the pipette each time you dip it in a new solution.
• Shake bottle for loosening.

First Slide

Note: A 1:1000 dilution of 1 micron beads is 1 part concentrated bead solution out of 1000 parts by volume of final bead water solution.

• When making a solution, DO NOT dip the pipette tip into one bead solution and then into another. Discard the tip EACH time you sample from a different bead solution as the bead solution is 3 to 4 orders of magnitude greater in value than the tips and cross contamination can make the experiment difficult to run! Once the solution is created in one of the small vials, shake it, then label the vial. Now assemble the slide. Each slide lasts only a few hours. It may be a good idea to label the tips for water, salt solution, etc. so that cross contamination is avoided. In addition, once a solution is made, it can be labelled and stored for the duration of the experiment.

Floating Bead Slide

To make a solution with floating beads you will use the distilled water to create a 1:3000 dilution of 10% by weight 2 micron beads, located in the fridge. Higher dilutions will be necessary for smaller beads and may be necessary for 2 micron beads. This will become evident when trapping of the beads is attempted. If the solution is too concentrated, it will be hard to find one bead alone to trap or, once one is trapped, there may be a constant barrage of other beads, knocking the original out of the trap. One way to create a 1:3000 dilution of 2 micron beads in distilled water is to use a two vial process. The two vial process also will make a 1:500 concentrated solution which will be used in the experiments as well. Locate the 2 micron bead solution found in the fridge. In a small plastic vial add 10 microliters of the concentrated bead solution to 5 mililiters of distilled water, creating a 1:500 dilution. Now using another small plastic vial, dilute about .5 mililiters of the first solution in more water, creating a 1:3000 dilution of beads to distilled water. If the pipette with the 1-10 microliter volume intake is available, this type of solution can be made directly in 1 vial. Once the solution is created in one of the small vials, shake it and label the vial. Now assemble the slide.

• How should the solution density change as a function of bead diameter to preserve the same bead to water concentration?

Stuck Bead Slide

To make a slide with stuck beads: use a ~1 M NaCl solution (provided) of 1:3000 dilution of 10% by weight 2 micron beads, located in the fridge. The best procedure to keep the beads from clumping is to first dilute the beads in a small amount of distilled water (maybe 500 microliters) and then add the 1 M NaCl solution. Higher dilutions will be necessary for smaller beads, as the solutions are all created by weight. For the stuck bead slide, it is best to let the slide rest for about 1/2 hour before using, to let the beads settle onto the surface of the glass.

Slide Assembly

To assemble the slide, first place (or reuse) a Kimwipe on the slide assembly table to create a clean place to set the slide. Any dirt or oil from your finger can obscure the light and/or scatter the laser beam, polluting the results of your measurement. Hence, keeping the center of the slide clean and the cover slip clean are important. Take out a microscope slide and place it on the clean surface. Place two rings of plastic binder paper reinforments directly on top of one another on the glass in the center of the slide to create a "well". Now, after gently shaking the dilute solution in the vial, deposit ~35 microliters in the well. Place a coverslip directly over this and depress the slip with another slide or something, but not your fingers, so that it is held in place by the adhesion properties of water.

Slide Loading

1. All slides should be prepared with enough solution to hold the cover slip in place.
2. Lower the objective lens, so that it will not be scratched (or even touched) by the microscope slide as it is being loaded.
3. Place one drop of the immersion oil on the lens using the dip stick in the bottle of oil. This should be 1 small drop, so let most of the oil run off of the dip stick back into the well before proceeding. Avoid touching the dip stick to the objective. Oil only needs to be added to the objective every other slide.
4. Place the slide into the slot on the stage with the cover slip down. This is an inverted microscope, after all!
5. Raise the objective until you see the oil make contact with the cover slip. This Z direction will have to be adjusted later to focus the microscope and laser. To be in focus, the Z position will usually be between +20 to -5 in the Z direction, once contact has been made.

First Trapped Bead

• To illuminate the beads, you must turn on the blue LED by flicking the switch on the silver box labeled "Microscope Illuminator" on the first shelf above the table. Without adjusting it, it should be set to around 2.9 V, which you can check with the multimeter immediately to the left of it. You should see an eerie blue glow emitted from above the stage.

Once you have read through all of the wiki previous to this, you can begin trapping beads. After loading the slide, turn on the Prosilica Viewer program from the desktop, which will allow you to see the beads. Now adust the objective slowly with the Z positioning knob until you can see floating beads in the Prosilica Viewer window. The objective lens should be touching the slide, with the layer of oil in between. If you advance too far you will lift the slide off of the stage, too little and you will not see the beads. Refer to the laser operation section and, following all safety procedures, turn on the laser. Once you have turned up the power to the lasing power, you will be able to see the laser reflected off of the beads you trap or any glass you are focused on. To find the edges of the well, first retract the objective in the Z direction until you see the laser reflected in a bright circular pattern on the near coverslip. To find the far side, advance the laser past the floating beads until you see the circular diffraction pattern on the microscope slide. The laser is in best focus just past the coverslip away from the microscope slide. The easiest way to trap a bead is to find one near the coverslip that is not stuck, focus just past it and then move the center of the trap over the bead. When the bead is trapped it will reflect the laser into the camera. If the bead is not stuck to the glass, you should be able to move it in the X and Y directions and then, very carefully, advance the bead away from the coverslip by advancing the objective by moving in the Z direction. If the bead falls out of the trap, try again with another bead near the coverslip. Trapping beads successfully away from the glass may take some practice.

The Optical Trapping Program

Before, we used Prosilica viewer to view and trap our first beads. It is not neccessary to use this program after this initial round of practice, since the People's Optical Trapping Suite program has a camera included that can be used to see and capture beads. The People's Optical Trapping Suite Program can be found under the folder: C:\Optical Trapping Code\BrownianApplication\bin\Release Double click BrownianApplication.exe. (Not BrownianApplication.vshost.exe) DO NOT EDIT THE PROGRAM! Ignore the red exclamation points on the folders. 5 windows will open.

• The Stage Control window controls the picomotors, which move the stage in X and Y directions, remotely. These can be also be controlled using the joystick or by turning the X and Y knobs manually.
  • To use the joystick, press the middle button on the joystick labeled "Driver" once. This sets the X and Y axes appropriately. To switch between step mode and joystick mode, use the control window. Be aware that in switching back to "joystick mode" from "Step Mode" you must also press the "JON" button to activate the joystick. The joystick should vary the frequency of the picomotor steps depending on how far you push it to one side. It will not go diagonal.
  • To use the program directly, select the "steps mode", set the frequency and number of steps (by typing into the text boxes and then hitting the "set" button). Now when you press Up, Down, Left and Right on the stage control the stage will be moved at this preset direction. You can hit "stop" to abort the current movement. If the stage control or joystick ever malfunction, the program may need to be restarted. Sometimes the power to the joystick may also need to be cycled. Turn off the laser first, if it is already on, then flip the power switch on the box marked "QPD and Joystick Box" to the off position. Please wait 15 seconds or so before turning it back on, as cycling power too quickly will cause the joystick not to work.
• The QPD Readout window is responsible for writing and reading all QPD signals. With this window you can view the average position of the particle, PSD data and X and Y positional data in real time. You can also record at different frequencies and for different periods of time, from 1/6 second at 12kHz to 4/3 seconds at 12kHz. It should be noted that the 8 second 1kHz write button also produces a down sampled version of this data at 30Hz. The data produced by these is then written to a notepad file and stored in the specified directory. You should create a new folder in the "C:\OTZ Student Data" directory, and save all your data to that folder.
• The Control Window is the same as is used in the Brownian Motion lab. This is the particle tracking software. The "Camera Go" should produce the same image in the selection box in the Pass Through window. By checking find particles and track particles the program will subtract the nonmoving background and began tracking the particles as they move. These particle tracks can also be saved to a notepad file under your "My Documents" folder using the save particle tracks button. The minimum bead size parameters can be adjusted to track the beads better.
• The Pass Through Window contains the selection box and shows the microscope image in real time. Using the mouse, you can move the red selection box around. "Shift" + "Click" will resize the box. The selection box determines what area the particle tracking occurs in. You will likeley need to select a large area to see particles move longer distances.
• The Images Window will show the image of the area within the selection box before and after processing. In the post processing window, blue circles will follow the particles once they are found and red circles will follow the particles being tracked.

QPD Centering

Before any data from the QPD can be used, the QPD signal must be centered, which corresponds to the laser hitting the center of the QPD. Once you have opened the program, have turned on the LED and QPD, and are running the laser in an area of a slide free of beads, you can center the QPD signal in the QPD readout window.

The QPD position signal is the yellow circle on the QPD readout window which can be accessed by running The People's Optical Trapping Suite and using the QPD position readout window. This process will need to be used before trapping any beads if the signal is not already centered. Most of the time only the fine adjust knobs will be needed to recenter the QPD if it gets off center. Check to see if the signal is indeed centered. If the laser is not on or is not incident upon the QPD, it will appear centered, though small vibrations or adjustments will cause no change in the signal.

1. First make sure the blue LED light circle is centered around the black circle made by the objective contacting the slide.
2. Now approximately center the QPD readout signal using 2 rough adjustments knobs attatched to the condenser. The condenser, which has a swivel wheel also attatched to it, should be left around 0.6. When centered the rough adjustments will move at about 120 degrees to one another and the QPD signal will swing through the center of the graph along both directions.
3. Finally, use the fine adjustments located on the casing of the QPD to center the signal. These adjustments should move the QPD signal at roughly perpendicular angles to one another. The X and Y positions can be read off more precisely using the Vx and Vy position graphs in the same window.
4. If this was all done without a bead becoming stuck in the trap, then the QPD has been successfully centered. Any slight bump of the QPD will throw off this alignment so be careful!

• If a bead is trapped during this procedure, you must move to another region of the slide and repeat. If this is very hard to do, then your bead to water concentration is too high.

Part I. Calibration

Microscope Calibration

Pixel to Meter Conversion

A micrometer slide with markings at 10 micron increments was used to measure the pixels/micron and it was found that . Use this for all future calibrations involving pixel to distance conversion.

Position v. Step Number

Picomotor Step Size

To find the average step size in +Y,-Y,+X,-X directions, stuck beads must be used. A bead stuck to the glass needs to be scanned though the view plane using the Stage Control GUI at a controlled rate for hundreds of steps, taking a picture using the Prosilica viewer snapshot option before and after the controlled stage movement. The picomotors have about a 30nm/step movement, but this movement varies wildly at the beginning of a controlled sequence of steps and smooths out after the first 20 or so steps. Hence, the longer the distance the bead travels, the more this initial variation is averaged out over the number of steps. The pictures can then be analyzed in a program which counts pixels to see how many pixels the bead travelled. Then using the pixel to micron conversion and the number of steps taken, an average step size can be calculated. This procedure has already been performed. The diagram of step sizes at different frequencies appears at right. However, due to the fact that picomotors fatigue, it may be a good idea to check a few directions at certain frequencies to see if the chart is correct.

Note from Dr. Ayars: I've found that the Brownian Motion program can be used to track the particles while you're moving the stage, and from this you can get the picomotor stepsize quite easily. I did not find that this stepsize varied with step frequency once the initial "warm-up steps" had been taken. The stepsize did vary with direction, since the picomotor was working either with or against various translation-stage springs depending on direction. My tests (9/2009) came up with consistent results of:

left: 34.55 nm/step

right: 23.79 nm/step

up: 34.23 nm/step

down: 25.90 nm/step

I don't know how these values will change with time.

Position Calibration of QPD

To use the QPD, we must first measure its response to small movements of the bead around the center of the trap. This is known as a sensitivity measurement and will give us a conversion between the QPD voltage response and the actual distance travelled by the bead. As all of our force estimates depend on knowing the sensitivity of the trap, a good estimate of this is important and should be done at 2 sizes: 1 and 2 micron beads.

The primary method by which the sensitivity is found should be the Power Spectral Density Line Fitting. This is by far the easiest and most consistently reliable method for finding the sensitivity. If the Stuck Bead Scanning Method can be done well, this should give the most accurate data. The Peak Value Method should only be used, if time permits, as a check on the other two methods. It is advised that the first two methods listed above serve as an estimate for the trap sensitivity.

Power Spectral Density Line Fitting

The easiest way to find the sensitivity, ρ, is to plot the PSD data obtained by trapping a floating bead in water. Once a 4/3 seconds of PSD data has been taken from a trapped floating bead, this data needs to be transformed into a line. Determine the sensitivity from the slope of this line. Transforming the PSD data into a line is described by the Rolloff Frequency Method.

Rolloff Frequency Method

The 4/3 second 12kHz PSD data button will create a .txt file with PSDx and PSDy data. PSD data should be taken at several laser powers (3 or 4) that cover the range of laser powers from 130 mA to 285 mA delivered to the Laser Diode. The Power Spectrum Density (PSD) of a trapped bead has a Lorentzian profile described by: in Volts² / Hertz. Where β = 3πηd is the drag, η is the fluid viscosity of water, d is the diameter of the bead, ρ is the sensitivity of the trap, and f is the frequency of bead vibrations. If the fluid is water then we can take: η = 8.90 * 10 ^{- 4}Pa * s. Using this, a curve can be fit to the log of the data sets giving the rolloff frequency f_o. In addition the rolloff frequency relates to the relaxation time: .

• It should be noted that the notepad file with the PSD data in it gives the PSD data for X and Y directions at the n^th reading. As the n^th reading is NOT the frequency, it should be transformed into the frequency: .
• In addition, because the PSD is a result of a Discrete Fourier Transform, the data is symmetric and hence the upper half of the data should be discarded. To do this in Matlab, make a new matrix and specify which rows and columns of the old matrix you would like to keep, in the form "datanew=data(A:B,C:D)". This will make a new matrix called "datanew" that has only rows A through B and columns C through D of the old matrix.

Using Matlab

One good method to do this would be to use program which minimizes the least squares distance of every point from the predicted curve, giving two parameters of the equation, alpha and f_o. The Lorentzian can be recast as the equation: . Using properties of logarithms and the two Lorentzian equations above, it should be possible to solve for sensitiviy.

A number of programs have been written for you to use, both here and in other calibration methods. In order to use them, set your Matlab "working directory" to C:\OTZ Data\2009 Data\Matlab. You should see a number of .m files. The function we will use here is called fitlogcurve. As the curve minimization uses a random seed point in the data to start the minimization, the program should be run several times, until consistent results are seen and the curve roughly fits the data.

To use Matlab, you will have to import the data of the notepad file into a matrix. Then, create a vector with half of the PSDx points, one with half the PSDy points, and one with the frequency. Take the log of these data sets, and plot log(PSD) vs. log(freq) for both PSDx and PSDy. The plot should look like the Power Spectrum Density as it appears in the QPD readout window.

The program then takes these two vectors as inputs and fits the theoretical curve to them. The curve will be plotted next to the data and the parameters α and f_o given. This is the function to call in Matlab:

[estimates, model] = fitlogcurve(logfrequency, logPSD);

Where model is a function which takes in estimates to produce the theoretical curve:

[sse, FittedCurve] = model(estimates);

You do NOT need to call the function "model" however. It is done for you in the function "fitlogcurve".

Stuck Bead Scanning Method

This process can be rather difficult and time consuming if done as the sole method for finding the sensitivity of the trap. For this reason, it is suggested that you spend no more than 2 days in lab working through this and that you view your answers as only approximate. The conversion this gives you should be compared to that obtained from fitting a line to the PSD data, described above.

The QPD output is roughly linear at +/-250nm around its center of detection, though this varies with bead size. For 5 micron beads there may be almost no linear region. The scattering of the laser changes with the size of the bead and hence, it is important to use several bead sizes when calibrating (I suggest 1 and 2 microns). The idea behind calibrating the QPD is to scan several bead sizes stuck to the microscope slide through the center of the laser at a controlled rate. This will give us a conversion between the QPD voltage output in X and Y directions to actual distance travelled. The picomotors produce an average step size of around 30nm depending on the direction of movement. However, this step size varies largely when beginning a movement and has a small hysterisis effect which causes a small drift in the direction perpendicular to the intended movement direction. Keep this in mind when performing this calibration. We will move a bead through the (0,0) position of the laser trap along the X and Y axes.

A graph above shows the X voltage responses plotted with arbitrary units. Note the green linear region at which point a line is fitted.

1. Using a stuck bead slide, in an area of the slide lacking beads, turn on the laser. If no beads are sucked into the trap, use this area to center the QPD signal.
2. Once this is done, center a stuck bead in the laser trap. This can be rather difficult as the QPD signal reads Vx=0,Vy=0 whenever the bead is far outside of the trap. The bead is actually centered in the trap when the QPD signal response is the greatest for small movenments (10-50 steps) in the X and Y directions and QPD signal is at (0,0).
3. Once the bead is centered, move the bead away from the trap center in the -X direction (direction is arbitrary) just far enough so the laser is not scattered by the bead. Stop the QPD readout.
4. Set the stage control window to move the bead in the +X direction twice as far (or more) as you moved the bead in the -X direction at a frequency of your choice. 100Hz movement seems to be a good pace as the stage vibration is not too bad.
5. The next two steps must be done in tandem:
   1. Click the Write 8s at 1kHz button and
   2. Began scanning the bead through the laser path (in +X direction) using the preset frequency.

This will take 8 seconds of QPD voltage data, sampling at 1kHz and record the QPD signal for the controlled step frequency. The readout of the QPD is saved as a .txt file to the directory indicated and labeled in sequential order. A graph of the QPD readout will be shown after the 8s period on the Vx and Vy versus time graphs on the QPD readout window. The graphs should both read roughly 0 at the very beginning and end of the run and you should see a signal like that of the one pictured above in the direction in which you are scanning the bead. The perpendicular direction (in our case, Y) should have very little change and be roughly symmetric around the linear region of the graph of Vx versus time. If both of these occur, this may be a good data set. You can, however, scan slightly off of center and see what appears to be a "good" data set but is actually not. That is, one in which a linear region appears in the scanning direction, but a large change appears in the perpendicular direction. To avoid this, you may take several scans and select the best few for each direction and compare. Note: Due to the hysterisis effect of the picomotors, centering the bead and then not moving the bead too far from the center is very important when taking a scan.

This notepad file can then be used to produce a graph of the QPD readout relative to time in Matlab. Plot both Vx and Vy verses time and note the time points that bound the linear region. There are two useful linear fitting programs in the same directory where fitlogcurve was located. They are called vxregression and vyregression. Open these programs in the editor and examine them. It should be fairly easy to understand what they do. Next, call them in MatLab using the appropriate inputs.

• Why does the data seem to "pulse"? (Hint: look at the frequency of the larger pulses.)

Once you have fitted a line to the linear region, estimate the QPD voltage response to bead distance traveled. Using the step size for a given direction, we can find a conversion between QPD voltage response and actual distance moved by the bead.

After recentering, repeat this meaurement for the perpendicular (Y) direction. The sensitivity in the X and Y directions may be quite different. For this reason, it is important to take several bead scans in each direction and look for some consistency. This method should then be checked against the other methods of finding the sensitivity of the trap. You should do this for at least four power levels in X and Y (for a total of 8 data sets) that span the allowed range of the laser (such as at 10, 20, 30 and 40mW). Note: Because the laser passes through numerous optical devices before it reaches the slide, the actual power applied in the trap is different from that displayed on the Laser Diode Controller. For your convenience a conversion table has been provided to the right, with the reading on the LDC in the ILD mode.

Once you have calculated the sensitivities for all four power levels in X and Y, plot them verses laser power in Matlab.

• Is the relationship linear?

Power Spectral Density (PSD) Peak Value Method

This procedure requires a floating bead in deionized water. Though this calibration lacks accuracy, it is very fast and easy. It cannot be used as the sole source for the sensitivity of the trap. For this procedure, you will need to trap a bead in the laser. Once this is done, you can simply read off the Peak Value, P^v, of the Power Spectral Density * frequency squared (PSD*f^2) graph. This is the height of the graph at the plateau of the PSD*f^2 graph. This can be used to find a rough estimate for the voltage to displacement calibration using the equation for the sensitivity in [Volts/meter]: Where P^v is the peak value. This assumes the fluid is water.

Trapping Force/Stiffness Calibration

The optical trap exerts a restoring force, modelled by Hooke's Law: F= -k*x,which holds the beads in equilibrium in the x-y plane. There are 3 methods to measure the trap stiffness, k. The first two methods use data collected from the free-floating bead slide and should produce similar results. These two methods are the Thermal Excitation Method and the Power Spectrum Density Rolloff Frequency Method. The final method uses the properties of a Newtonian Fluid and models the force exerted by moving fluid on a particle using Stoke's Drag relation.

Equipartition Method

This method is based on the equipartition theorem, which states that each degree of freedom in a harmonic potential has of energy. Using this theory, we can relate the thermal energy of a system to the instantaneous displacement of small particles via the statistical mechanical idea of heat as: , where < x > denotes the average over x, the position of the particle along the x direction. So, a floating bead must be trapped and the QPD voltage data, Vx and Vy, information taken for at least 4 different power levels (again, 10mW-40mW recommended). This voltage data must then be converted into the actual position via sensitivity. A smaller bead, around 1 or 2 microns, vibrates more and is thus, more suitable for this procedure. This can be done using the QPD Position Readout control in the main program by taking an 8 second reading of the data. It may be useful to note the equivalent statement: . Thus, we can find an estimate of the stiffness of the trap in the x and y directions:

The observations used in this analysis are assumed to be independent. Hence, for the calculation to make sense, the particle positions must be taken at intervals greater than the relaxation time, > > τ_o, which can be derived from the rolloff frequency, of the trap. As the relaxation time of the trap should be around 10ms, to insure independence, the data points should be taken at around 30Hz. Since the slowest sampling rate available is 1kHz, the data must down sampled, to avoid correlation between successive data points. This has been done automatically and two of the columns in the 8 second, 1kHz notepad file are downsampled to approximately 30Hz, giving around 240 independent data points. Due to the random nature of the down sampling, each of these down sampled columns may end with a string of 0's. These should be removed. Because this is a somewhat small amount of data, it may be a good idea to take several 8 second readings and compare the different k values.

• Why might there be a difference in ?
• What must be assumed for the equipartition theory to work (e.g. about the potential of the trap)?
• Why must the successive data points be independent?
• How does this idea of autocorrelation between successive values of the position, x, relate to the Power Spectrum Density?
• BONUS: How can this independence be tested? i.e. How can the correlation be measured?
  • To understand how to test this it may help to consult Reif and look into the ideas of covariance and autocorrelation. If time permits and you are interested in this idea, by taking several data sets (maybe 100 of 1 second each), each giving a value: k_i, you should see some variance in k. You can measure the mean and standard deviation in k using 2 different sampling rates (one above and one below the correlation time of the trap) and then measure the autocorrelation.

Using Matlab

The program Equipartition.m should be found in the same directory with all the other Matlab programs. This program takes six inputs: the 4 data sets you collected for the different power levels, a vector of your x sensitivities, and a vector of your y sensitivities (calculated earlier from the rolloff frequency/stuck bead method). It will graph the stiffness in both X and Y and perform a linear fit. A quick glance at the program in the Matlab editor should be sufficient to understand how it operates.

Rolloff Frequency Method

We have already used the Rolloff Frequency Method to find trap sensitivity using the PSD data, but we can also use it to find the trap stiffness, k. More specifically, the rolloff frequency parameter found from fitlogcurve gives the trap stiffness from the following relation:

k = 2πf_oβ again where β = 3πηd is the drag, η = 8.90 * 10 ^{- 4}Pa * s is the viscosity of water, and d is the diameter of the bead.

Compare this value with that obtained from the Equipartition method. Which is more accurate? Consult a GSI for the actual value.

Stoke's Drag Method

DO NOT DO at this point - correct spring constant of stage first

This method uses the force created by a laminar flow of fluid past a spherical bead, given as: F = βv, where β = 3πηd as given above, and v is the velocity of the water. This is used to estimate the restoring force of the trap. The idea is to induce a constant laminar flow, and by estimating the velocity of the flow when the force of the water is in equilibrium with that of the trap, predict the force on the bead.

One way to create a laminar flow is by making a flow cell. This can be done by using about 3 layers of double sided tape on either side of a microscope slide and a coverslip on top to create a small rectangular flow cell, as in the picture. The coverslip in the picture, show in grey, is much larger than the coverslip provided. Set up this flow cell, fill with a 1:500 part 2 micron bead to water solution and place on the stage with the coverslip down. Using the Prosilica Viewer, the beads can be seen in suspension. Using an approximately 1x2" piece of Kimwipe, a corner can be rolled into a small point. Care must be taken to make several of these points the same. The thickness of the tip and the areas near the tip are the most critical in determining flow rate. Some will be used to estimate the flow velocity of the water, while other similar ones will be used to induce this flow once a bead is trapped. Th point of the Kimwipe can then be touched to the edge of the water solution in the flow cell to induce a laminar flow of the beads near the center of the flow cell. Near the edges of the flow cell, the viscosity of the water will change the flow. The velocity must be estimated from the rate the beads move at, once a steady flow has been established. There should be a brief acceleration of the beads before an approximately steady flow rate is established. Due to the rapid rate of flow of the beads, it will be necessary for one person to start the flow of solution and another to watch the viewing screen and estimate how long it takes for a bead to go across one entire screen. This can be done with a stop watch. The full viewing screen is 659 x 493 pixels. This can be converted to about 44.5 x 33.3 micrometers using the pixel to distance conversion. After every trial of sucking up water using capillary action of the Kimwipe, more solution must be added to the flow cell to maintain the same initial set up. Once the velocity of the flow near the center of the cell has been established, a bead can be trapped near the glass and moved to the center of the trap. Using the QPD position readout GUI, an 8 second, 1kHz sample can be taken of the bead while a laminar flow is induced around it. This data should show a part where the bead is trapped and in constrained brownian motion and another part where the bead has been shifted to the side, due to the force of the water flow. If a constant laminar flow is established, the two forces will be at equilibrium for some period, which should show up in the positional data. The average deviation from the center of the trap due to laminar flow can then be estimated using a graphing program like Excel. This will give an average deviation, <x> and knowing the force on the bead induced by the flow of water past it, one can estimate the trap stiffness, k. Due to the difficulty in inducing a laminar flow using the Kimwipe, it is suggested that the team take several data sets (atleast 5), each time refilling the flow cell to restore original conditions.

At right is an example of a data set of 8 seconds, sampled at 1 kHz, where laminar flow was induced using the flow cell. It can be seen that the flow becomes approximately steady between 2 and 4.7 seconds in the Y direction. Here the force of the trap and the laminar flow force came into equilibrium. The bead remained in this position, vibrating around this point due to the normal Brownian motion. After that, the Kimwipe was removed and the bead returned to normal Brownian motion centered in the trap.

This method is not to be used alone because the difficulty of the set up can cause large errors. It does, however, have the advantage of using a completely different set of physical data to estimate the trap stiffness than the previous two estimations. It should also be noted that the water will flow differently if near any edge of the flow cell due to interactions of the water with the edges. The solution can be modelled as an incompressible Newtonian fluid just as water. A discussion on sources of error, assumptions and the physics used to estimate the flow of water is necessary here.

Another way to see the drag effect on the bead, is to first trap a bead in a dilute (1:3000 or so) solution prepared in the normal "well" fashion and move it away from the cover slip. Then with some practice, you can learn to turn the knob with a regular speed. Estimate the rate you are turning the knob using stuck beads and a stop watch. If you start an 8 second 1 kHz reading and began the movement slightly after starting, the program will graph the positional data from the QPD. If a smooth turn can be accomplished, you should see data similar to that of the flow cell shown above. I found it hard to avoid accelerating the knob rotation speed because the static friction presents a significant barrier at the beginning of the rotation. For this reason, the data is often irregular. However, this is a fast and easy way to see the effect, though useful data is hard to obtain from this method.

An important question to ask is: Could the stepper motor be used create a laminar flow of water past the bead? Why or why not?

Part II. Investigating Flagellar Locomotion in E. Coli

Escherichia coli is a rod-shaped bacterium about 1 μ in diameter by 2 μ long. The cell is propelled by a set of four helical flagellar filaments that arise from separate positions on the cell surface but intertwine to appear as a single flagellum in forward movement. Each filament is driven by a rotary motor at its base that turns counterclockwise (CCW) in forward swimming. A cell swims steadily forward in a direction parallel to the long axis of the cell with the flagellum "pushing" for a second or two, called "running". Then it moves erratically in place for a shorter time, called "tumbling", then runs off in a new direction. Tumbling behavior is caused by one or more of the flagellar filaments turning clockwise (CW), during which time the flagellar filaments separate. You can see the run and tumble pattern of behavior in this movie of fluorescently labelled E. coli.

In this section of the lab you will trap E. Coli bacteria which move using flagella. First take a sample of the liquid in your mouth. You can minimize the air bubbles by dragging a wood tongue scraper against the inside of your mouth. Load a slide with the saliva using the small coverslip (about 1" square). You may need to dilute the solution if it is dense in motile bacteria. E. Coli look like long cylindrical tubes. This part of the lab will only give very approximate estimations of force due to the odd shape of the bacteria. So anything within an order of magnitude should be considered "equivalent."

• How does the shape of the bacteria affect the laser trapping?

Stoke's Drag Flagellar force estimate

Using the particle tracker, try to track some of the E. Coli swimming by and estimate their velocity. Using Stoke's Drag, you can estimate the force they propel themselves through the water with. The Stoke's Drag force of a laminar flow past a spherical bead of diameter d is: F = βv, where β = 3πηd, v is the velocity of the water flow past the object, and where η is the fluid viscosity, which we may take to be water: η = 8.90 * 10 ^{- 4}Pa * s.

It may also be useful to estimate the velocity of the bacteria with a stopwatch by counting the number of pixels it travels in a given time. This can be a check on the particle tracking velocity since it may be difficult to determine which particle the E. Coli is from the recorded particle tracking file.

Bacteria Trapping

Swimming Force Estimate

First find a highly motile E. Coli. With a somewhat lower trapping force, trap it while it is swimming. When trapped strongly, the bacterium should be lengthwise, with the long portion of its body aligned along the direction of the laser. In this position the bacterium is completely confined and usually cannot swim out of the trap. Allow the trapping force to decrease so that the bacterium is swimming around in the trap, yet confined. An estimate on its flagellar propulsion force can be made by slowly decreasing the laser trapping force until the bacterium can break free.

• An estimate of the propulsion force here is only meaningful if:

1. the bacteria swims out of the trap and does not just drift out of the trap slowly, by Brownian Motion.
2. the laser power when the bacteria swims out is above the lasing point, of course.

The trap stiffness, k, should scale linearly with laser power. Using your previous estimates of k at different power levels, extrapolate this data to estimate the force that the bacterium overcame when it escaped.

Observe the Tumble

Another important phenomenon to observe of the E. Coli is the tumble. When trapped at a low trapping strength, the bacterium should be moving around in the trap and not confined perfectly vertically at all times. With a bacterium that is particularly motile, you should be able to find this point by adjusting the laser power. The bacterium will try and swim out of the trap periodically and then relax back into Brownian motion, where it may become temporarily vertical. This periodic relaxation of the confined bacteria is known as the tumble. At this point the bacterium is relaxing its flagellar bundle, then recoiling it and attempting to swim in a direction of less resistance.

1. Record an 8 second period of positional data of this bacterium in which you see periodic struggles and relaxations.
2. Find how far out of the center of the trap the bacterium swims in these periods of activity without quite freeing itself. This is r_max.
3. Knowing the trapping stiffness from previous experiments, we can estimate the swimming force of the bacterium by assuming the trap and the bacterium are in equilibrium. Then .

• To find you will have to average over the x and y position for a short period when the bacterium is fighting the trap in one direction. Averaging will help elliminate noise, which would skew the position. Since we didn't measure k in general, but for each direction x and y separately, estimate a maximum and minimum force that the bacteria is pushing with.
• Record this in the lab report and show examples of the tumble.

Part III. Investigating Internal Transport in Onion Cells

To begin, you must first have an onion you provide and a bag to hold it after you cut it open in the lab. First, read through the "Analyzing Intracellular Movement in Onion Cells" under the Brownian Motion in Cells page. Our goal in this section is to analyze the force exerted by Myosin to move the granules in the onion cell. These small granules (or vesicles) are about 1 micron in diameter.

• NOTE: The saline solution may be in the cupboard above the BMC lab area.

Estimate Trap Stiffness on Granules in Onion Cell

Though the trap stiffness has already been classified for various beads in a water solution, the inside of an onion cell is very different. The viscosity of the solution is different and the granules have different indexes of refraction. For this reason, the trap should have a different trapping strength on the granules. One way to estimate the trap stiffness on a granule is to find one that is not being transported and appears to be in Brownian motion. First, take a snapshop of this granule and estimate its' size.

• Trap this bead and use the equipartition measurement to find the trap stiffness. This requires that a micron bead, which is about the same size as the granule, has been scanned through the trap to give the QPD voltage to distance conversion.
• Or, if you can find an estimate of the viscosity inside an onion cell, you can take a Power Spectrum Density graph of the granule and estimate the stiffness this way. The onion cell is mostly water, but more viscous, so its' viscosity is about 3 to 6 times that of water.

Estimate Cellular Transport Force

One of the particles responsible for transport is Myosin. This acts as the transport mechanism which binds to a "cargo" molecule and to an Actin strand. We would like to estimate the force used by this mechanism to transport the granules seen in the onion cell. Once you have looked over the onion cell and found an area of regular movement, you can use the trap to estimate with what force these granules are being moved with. These transport processes may be somewhat hard to find given this microscope setup has a very small of depth of field and very high magnification (around 2000x). When using the trap, the beam is best focused near the center of the cell, where a circular diffraction pattern can be seen. One method to do this is to find one of the endoplasmic reticulum or "tubes" that is in your field of view that you see the granules traveling in. Find a narrow "bottleneck" on this tube where the granules are passing through at about one at a time. These next few steps must be done in quick succession:

1. Turn on the laser,
2. Start the 8 second reading just before the granule moves into the trap.
3. The granule in active transport should be caught in the trap.
4. Once the reading has been taken you should see that the Myosin appear to be "pushing" the granule regularly, as if trying to overcome the restoring force of the trap. Look at the graph of position versus time for evidence of this pushing and record the laser power.

• Eight seconds of positional data will give an idea of how the Myosin push on these granules. Is this a regular force? Do they continue to push once trapped? What conclusions can you draw based on these experiments? How strong do the Myosin appear to be pushing on the trap? \
  • Include a graph of the force versus time developed by the Myosin in the report.
• If the trap strength is slowly decreased and the granule breaks free, then based on the trap stiffness given a certain laser power, the maximum force of the Myosin can be estimated. Since the pushing is irregular, the power will have to be lowered periodically, until the granule is pushed out of the trap. What is this maximum force?
• Using the particle tracker, find a particle which has been tracked well and during a period of regular movement. Using Stoke's Drag equations, estimate the force applied to the granule in order to overcome the drag force of the fluid in the cell.

How do these methods compare? How well does the trap work in the cell as opposed to water?