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MODULE 6: OPTICAL TRAPPING

The previous lab modules have introduced methods of seeing the micro- and nanoscopic world,

and this lab introduces means for directly interacting with it by applying force using an optical trap.

You will build a single-beam gradient force optical trap on top of the brightfield microscope

constructed in Module 4. Brightfield imaging will serve as the detection mode in this module. You

will see the Brownian motion microscopic particles undergo and produce optical forces of sufficient

strength to overcome these thermal fluctuations and trap the particle. Introductions to optical

trapping technology can be found in several reviews.†

I. Introduction

A. Radiation Pressure

Working at Bell Laboratories in 1970, Arthur Ashkin discovered that micron-sized particles could

be trapped using only the radiation pressure from laser light.‡ Despite the fact that a photon has no

rest mass, it does carry momentum according to De Broglie’s relation, A single photon carries

momentum of l=h/p, where l is the wavelength and h is Planck’s constant. On the time and

distance scales we experience in daily life, this momentum is negligible. For example, if all 800 mW

from the white LED used in the previous modules reflected off an apple for one second, it would

transfer 5.3 x 10-9 kg*m/s of momentum,§ which is sufficient to accelerate an apple (~0.1 kg) from

rest to 53 nm/s. However, microscopic particles, such as a 1 μm micron diameter glass bead or a

single cell bacterium (mass scale ~1 pg), would be accelerated to 5.3 km/s in 1 ms. In vacuum this

may be the only force experienced by a particle, but in solution the thermal energy carried by

solvent molecules imparts nearly random velocities to a solute causing it to diffuse randomly –

Brownian motion. Even if most of the light is transmitted and in the presence of Brownian motion

and friction, this force is significant enough to be observable. † “Optical Trapping,” K.C. Neuman and S. M. Block, Rev. Sci. Instrum. 75, 2787 (2004) and Single Molecule Analysis [electronic resource]: Methods and Protocols, Chapter 1. ‡ “Acceleration and Trapping of Particles by Radiation Pressure,” A. Ashkin Phys. Rev. Lett. 24, 156 § Total energy E=Σ hc/λyields p=2*800mW*1s / c, where the factor of 2 arises from considering the momentum transferred during backscattering (complete recoil)

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B. Brownian Motion

In 1827, while Robert Brown was studying the fertilization process in flowers, he observed that

minute particles ejected from the pollen grains undergo “rapid oscillatory motion.”** He realized that

this effect was also observed by earlier microscopists such as Leeuwenhoek, and ruled out the

hypothesis that pollen grains were alive by showing that inorganic materials, such as glasses and

minerals, undergo the same motion. Further studies by G.L. Gouy†† demonstrated that Brownian

movement was more rapid for smaller particles, at higher temperatures, and in less viscous fluids.

This effect was independent of the composition or density of the particles. It was known from the

kinetic theory of gases that the root-mean-square velocity in each direction was given by the

Equipartition Theorem, 12𝑚 𝑥() 𝑡 =

12𝑚 𝑥)) 𝑡 =

12𝑚 𝑥,) 𝑡 =

12𝑘.𝑇,(1)

𝑥) 𝑡 =3𝑘.𝑇𝑚

,(2)

and so motion could not cease at finite temperature. In 1905, Albert Einstein used this to describe

this average properties of Brownian motion with a diffusion equation,‡‡ 𝜕𝜌(𝑥, 𝑡)𝜕𝑡

= 𝐷∇)𝜌 𝑥, 𝑡 (2)

where 𝜌(𝑥, 𝑡) is the probability of finding the particle at position 𝑥 at time 𝑡, and 𝐷 is the diffusion

coefficient. Einstein was able to derive the diffusion coefficient in terms of physical parameters,

𝐷 =𝑘.𝑇6𝜋𝜂𝑟

,(3)

(Measurement of this diffusion constant actually allowed consistent measurements of Avogadro’s

constant.) This theory predicts the mean position and velocity, 𝑥 𝑡 = 0, 4

𝑥 𝑡 = 0, 5

as well as the root-mean-square position,

𝑥) 𝑡 = 6𝐷𝑡, 6

** “A brief Account of Microscopical Observations made in the Months of June, July, and August, 1827, on the Particles contained in the Pollen of Plants; and on the general Existence of active Molecules in Organic and Inorganic Bodies”, R. Brown, Philosophical Magazine N. S. 4 (1828), 161–173. †† “Brownian movement and molecular reality,” J. Perrin, translated from the Annales de Chimie et de Physique, 8me Series, 1909, by F. Soddy, Taylor and Francis, London, 1910 ‡‡ “Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen”, Einstein, A., Annalen der Physik 322 (8): 549–560.

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It is illustrative to also consider the motion of one particular particle, which can be described by a

Langevin diffusion equation,§§

𝑥 𝑡 = −𝛾𝑚𝑥 𝑡 + 6𝐷𝑓 𝑡 , 7

where 𝛾 = 6𝜋𝜂𝑟. This equation shows that the acceleration of a particle, 𝑥 𝑡 , arises from the sum

of two opposing factors: a friction term, which is proportional to the velocity, EF𝑥 𝑡 , and a random

acceleration, 6𝐷𝑓 𝑡 , where 𝑓 𝑡 = 0 and 𝑓 𝑡( 𝑓 𝑡) = 𝛿(𝑡( − 𝑡)).

C. Optical Trapping

The considerations in Section IA describe the scattering forces that result in optical pushing.

Optical trapping requires both pushing and pulling type-forces to overcome Brownian motion and

constrain a particle to a fixed point, and this is largely due to refraction. Since light must be focused

in order for it to efficiently interact with microscopic particles, let us consider the forces induced on

a typical glass microsphere using ray tracing.

Figure 1 shows the resulting forces on a microscopic particle of higher refractive index than the

surrounding medium due to light refraction. In stable optical trapping, the light must produce

momentum that overcomes any random Brownian motion that may occur. Note that if the particle

undergoes a random fluctuation that sends it away from the focal point of the light, there will be a

§§ See for example, Lecture Notes CHEM 430b/530b: Langevin Equation (V. Batista)

Figure 1. Forces due to light refraction. Red lines show rays incident on a particle of higher refractive index than the surrounding medium to be trapped. Blue lines show the momentum transfer to the particle due to refraction. For weakly focused or collimated light (A) the net effect is to push the particle forward. In a beam with Gaussian mode profile (B), the particle is driven towards regions of higher intensity. For a focused beam (C and D), the wider angle rays pull the particle towards the source while the rays near the center (or from a lower numerical aperture lens) contribute little to the refractive force, but can still scatter (reflect) or be absorbed and contribute to heating.

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force that tends to push it back towards the focus. Optical traps typically result in harmonic

restoring potentials, with a stiffness 𝑘,

𝐹I 𝑡 = −𝑘I𝑥 𝑡 (8)

Note that this stiffness may be different in the x, y, and z directions. For example, an elliptical beam

may provide different restoring forces in the two transverse directions. Also, typically the intensity

gradient in focused light is less along the optical axis than in the transverse directions, and so the

restoring force out of the focal plane is typically 30% less. In this experiment, only the stiffnesses in

the transverse directions (𝑘I, 𝑘K) are calibrated.

When calculating the magnitudes of the forces exerted by an optical trap, two limits may apply: (1)

Rayleigh Scattering, when the wavelength of light is much larger than the diameter of the trapped

object (𝜆 ≫ 𝑑) or Mie Scattering, when the wavelength of light is much smaller than the diameter of

the trapped object (𝜆 ≪ 𝑑). Since our trapping laser is 980 nm and the microspheres and bacteria

trapped in this module are 1-5 μm in diameter, our situation is at the boundary of these regimes

and the curious reader is referred to accurate numerical calculations.***

D. Force Calibration A linear restoring force (Eq 8) results in a harmonic trapping potential,

𝑈(𝑥) =12𝑘I𝑥)(8)

In thermal equilibrium, a particle subject to this potential can be found at positions obeying a

Boltzmann probability distribution,

𝑃(𝑥) =𝑘I

2𝜋𝑘.𝑇𝑒STUIV/)TWX,(9)

where 𝑥 = 0is defined as the center of the trap. In kinetic theory, you may have been introduced to

the Maxwell-Boltzmann probability distribution for particle speeds, arises from the exact same

considerations for a free particle. Figure 2 shows this probability distribution overlaid on the

trapping potential for several trap stiffnesses and temperatures. At higher temperatures, the

particle will have sufficient energy to be found further from the center of the trap, yielding a wider

probability distribution. Similarly, the same effect may arise from a weaker force constant. By

*** “The effect of Mie resonances on trapping in optical tweezers,” A. Stilgoe, T. Nieminen, G. Knöner, N. Heckenberg, and H. Rubinsztein-Dunlop, Opt. Express 16, 15039-15051 (2008).

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assuming a temperature, one can measure the particle positions and estimate the force constants

for the optical trap, 𝑘I. Equivalently, the variance in particle positions is related to the stiffness as,

𝜎) = 𝑥 − 𝑥 ) =𝑘.𝑇𝑘I

,(9)

A useful conversion constant is 𝑘.𝑇 = 4.142𝑝𝑁 ∙ 𝑛𝑚 at 300 Kelvin. Because the variance in

particle position is related to the average thermal energy in this degree of freedom, Equation 9 is a

statement of the Equipartition Theorem and represents the Equipartition Method for stiffness

calibration. In addition to modeling the variance in particle position, the frequencies of its fluctuation

also report on the stiffness of the trap (see Neuman K.C., et al.†††)

††† Optical Trapping,” K.C. Neuman and S. M. Block, Rev. Sci. Instrum. 75, 2787 (2004)

Figure 2. Equilibrium probability distribution of particle positions overlaid with trapping potential at with varying force constant and temperature.

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E. The E. coli Flagellar Motor One of the first communities to use forces from optical traps to interact with the nanoscopic world

have been biophysicists and biophysical chemists studying intrinsically force-bearing and force-

generating proteins, such as the muscle protein titin‡‡‡ and motor proteins myosin, kinesin, dynei,

and ATP synthase.§§§

In this lab module, you will study the molecular motor responsible for E. coli motility. Figure 3C-E

show the architecture of this multiprotein assembly. The flagellar filament is a thin, 10 μm long

cork-screw shaped structure that protrudes from the motor protein assembly. The Motility proteins,

MotA and MotB, are transmembrane proteins that use a proton gradient to generate torque, and

sit surrounding the MS ring. In our assay, the optical trap is used to apply force to the tip of an E.

coli cell sufficient to stall the motility proteins (Figure 3A). This force generates a torque on the cell,

𝜏 = 𝑟×𝐹.(10)

At the stall point, the torque generated by the trap exactly balances the torque generated by the

motor, and the cell may escape the optical trap. The distance 𝑟 can be measured by observing

free rotation of the cell (Figure 3B). The escape force can be calculated by measuring the stiffness

of the trap, 𝑘, and the escape distance (𝐹abc = −𝑘𝑑abc). The escape distance can be estimated by

image analysis, and should be roughly 100nm.

The strain we will be using, RP437, was first used to study motility and chemotaxis in 1978.**** Our

samples were provided by the Emonet Lab at Yale, who use this strain to study how bacteria

sense and explore their environment.†††† The tethered assay was first developed in 1974,‡‡‡‡ when

Silverman and Simon discovered that a single flagellar filament could attach to a surface and result

in spinning of the cell body. It has since been used in a series of studies to quantify the energetics

of flagellar rotation.§§§§

‡‡‡ “Folding-Unfolding Transitions in Single Titin Molecules Characterized with Laser Tweezers,” Kellermayer, M.S.Z., Smith, S.B., Granzier, H.L., Bustamante, C., Science 276(5315) pp.1112-1116 (1997) §§§ “Probing the Force Generation and Stepping Behavior of Cytoplasmic Dynein,” Gennerich, A., Reck-Peterson, S. L., Single Molecule Analysis: Methods and Protocols, Methods in Molecular Biology, vol. 783, Chapter 4; “The Physics of Molecular Motors,” Bustamante C., Keller, D., Oster, G., Acc. Chem. Res 34, 412-420 (2001).**** “Complementation analysis and deletion mapping of Escherichia coli mutants defective in chemotaxis,“, Parkinson, J.S ,J. Bacteriol. 135(1):45-53 (1978). ††††More information available at the Emonet Lab Website‡‡‡‡ “Flagellar rotation and the mechanism of bacterial motility,” Silverman, M. and Simon, M., Nature 149, 73-74 (1974).§§§§ See publications by Berg H.C., such as “Energetics of flagellar rotation in bacteria,” Manson, M.D., Tedesco, P.M. and Berg, H.C., J. Mol. Biol. 138, 541-561 (1980).

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Figure 3. E. coli flagellar motor stall force assay. A) A cell is trapped near the tip to halt rotation. B) To calculate the torque, the rotational radius, r, and the displacement as escape, desc, must be estimated. C) Schematic of the bacterial flagellum as a mechanical device. The parts thought to be responsible for torque generation are shown in thicker lines. They lie inside the cell except for the studs, which cross the membrane and are thought to be anchored to the peptidoglycan layer (PG). The filament or propeller lies outside the cell’s inner (IM) and outer (OM) membranes and is about 10 mm long. (D) Image of the flagellum. The image is an average of about 100 individual electron micrographic images of a frozen-hydrated preparation of flagella. The bar corresponds to 25 nm. (E) A color-coded copy of a slice through the cylindrically averaged flagellar structure. FliF, which forms the M ring, S ring, and socket, is shown in green. FliG, shown in blue, is attached to the cytoplasmic face of the M ring. FliM and FliN, shown in yellow, are thought to make up the C ring. The L and P rings are shown in pink, and the rod and the short stretch of hook are gray. (Panels C, D, E reproduced from “The Turn of the Screw: The Bacterial Flagellar Motor”, DeRosier, D. J., Cell 93(1) p17-20 (1998) under the fair use license.)

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II. Instrument Layout

Figure 4 shows the instrument you will be building. In addition to the brightfield microscope, a

980nm laser (up to 200 mW) is incident on the back aperture of the objective. In contrast to the

configuration for wide-field imaging (fluorescence or brightfield), the trapping laser is not focused to

the back focal plane of the objective; the objective is used to tightly focus the collimated light. A

telescope is used to expand the beam size such that it overfills the objective. While some light that

is rejected at the aperture is wasted, the resulting configuration maximizes use of the numerical

aperture of the objective and the increased intensity at the edges of the objective lens contributes

to a stronger optical trap (Figure 1C).

The 980nm light is directed towards the objective with a hot mirror, which is another name for a

dichroic mirror that reflects the infrared. Trapping laser light that is backscattered is preferentially

rejected; visible light from the LED passes through. An additional hot mirror is used to further

reduce the backscattered 980nm light. Both of these hot mirrors will allow ~10% of the 980nm

light to pass through, so your trapping laser should also be visible on the CCD. (What would you

expect the image of the trapping laser to look like?)

As an additional control, telescope lens 2 is mounted on an x-y translation stage. This allows for

fine control of the trap position without changing the focusing. This may be used for fine

adjustment of the optical trap in addition to the sample holder translation stage.

Figure 4. Schematic of the optical trap and brightfield microscope. Inset indicates different beam paths.

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III. Additional Safety Precautions In this lab you will be using a Class 3B fiber-coupled laser emitting up to 200mW at 980 nm. The

optical fiber coupling the light from the laser diode to the exit aperture is somewhat fragile and may

break when bent sharply. Additionally, the power supply driving the laser is capable of outputting

enough power to damage the diode; although a soft limit has been preprogrammed, please mind

the driving voltage and please review the data sheet in lab to estimate the output power.

Since infrared light is invisible, the safety glasses that shield your eyes from stray laser light will not

affect your ability to align the beam. In lab you will find an infrared viewer card that can be used to

visualize the beam during alignment. Near-infrared lasers pose the largest dangers in terms of laser

safety because the beam cannot be seen. (Recall that you could see as little as ~0.1mW of the

520nm laser. One is unlikely to look directly at the laser light without knowing.) Therefore, when the

laser is on, everyone in the room must wear laser safety glasses – absolutely no exceptions. To

minimize the chance of exposure, please also remove all reflective accessories, such as rings and

watches, during the lab period.

Of course, all of the safety precautions for optics from Module 4 and 5 also apply here. Be extra

mindful to block the beam whenever inserting or removing optics from the beam path.

The E. coli cells used in this lab pose a BL1 hazard, which poses minimal threat to safety.

However, please always dispose of biological material in the designated waste container for

subsequent autoclaving and removal.

V. Day One – Build the Optical Trap Step 1. Align the tube lens and CCD camera as you did in Module 4.

Step 2. Insert the hot mirror behind the objective. Since no light is on the table, you can sight

directly down the optical axis to make sure that the dichroic mirror is situated at 45 degrees and is

aligned squarely in front of the CCD.

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Step 3. Insert the silver mirror as shown in Figure 2. Coarsely adjust its position and tilt. It

should be angled at 45 degrees and situated in line with the hot mirror.

Step 4. Before turning on the laser, think about its beam path and add beam blocks to

prevent any stray light (such as the small leak through the dichroic) from leaving the

optical breadboard.

Step 5. Turn the laser on and adjust it to a low power. For alignment, you want the power to

be minimally visible on the alignment card. Check the manufacturer’s sheet to see which current

this corresponds to. To turn the laser on:

a) Switch on the QCombo-6305 power supply***** b) Turn the “Laser Enable” key to the “On” position c) Switch on the TEC (thermoelectric cooler) d) Verify that the driving current is 0mA and switch on the laser. e) Gradually increase the driving current until you can visualize the green laser spot on a

white card. Be careful as this step can result in stray reflections. Consider that any surface that the laser strikes can partially reflect the light and act as a weak mirror.

f) Always the alignment card to see exactly where the laser is at all times and make use of the beam blocks.

Step 6. Insert telescope lens 1 such that the laser is not displaced when moving it.

Visualizing the beam after this lens, you will see it diverge. Note that the center of the beam should

be fixed; this can be adjusted with the laser mount and its tip/tilt alignment screws.

Step 7. Insert telescope lens 2 such that the laser is not displaced when moving it.

Visualizing the beam after this lens, you should see a collimated beam with the same center at any

point.

Step 8. Prepare an iris to the exact height of the front aperture of the objective. Put an iris

on the rail right in front of the objective where the sample would be. When you close the iris all the

way down, you should still see the small front aperture of the objective. Note that the beam before

the silver mirror need not necessarily correspond to the position of this iris.

***** Instrument manual available at the QPhotonics website.

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Step 9. Use the irises you’ve prepared to adjust the silver mirror and hot mirror such that

the optical axis passes through the center of the objective. This is an identical procedure to

that shown in Module 5, Figure 7.

Step 10. Build the remainder of the brightfield microscope as in Module 4. Note that this

objective lens is on a stable 1.5” pedestal mount that allows for fine translation along two axes.

Step 11. Insert the grating test sample into the sample holder and do a “dry run” of how

you will bring the slide towards the objective without adding immersion oil. (That is, verify

that you can freely move the slide holder and no obstructions are in place.) Note where the

objective will meet the sample. Also note the surface area of the face of the objective where it will

meet the sample – this indicates how much immersion oil needs to be added.

Step 12. Add a drop of immersion oil on the slide directly above the location where the

objective will meet the sample and as it slowly drips down, move the slide in place to

meet the objective. Some tips:

a) Be mindful of immersion oil drips and spills, such as off of the applicator after you’ve added a drop onto the slide, by oil spots on your gloves, or by tilting the bottle. Clean any excess immersion oil off of non-optical surfaces with isopropanol and Kimwipes.

b) Optical surfaces should be cleaned of immersion oil using lens paper only. c) Minimizing dripping by wiping the applicator after dipping it into the oil. d) Practice may be required to calmly move the sample towards the objective until

full contact is made with the oil droplet without directly crashing the objective into the slide. Feel free to wipe away the oil and begin again.

e) Due to the viscosity of the immersion oil, you should not move the slide position as quickly as you did in Module 4; moving the slide quickly will result in air gaps between the slide and objective.

Step 13. Once the objective and slide are in immersion oil contact, you may adjust the

focusing using the fine threaded knob on the objective lens mount.

Step 14. Using the grating test sample, derive a pixel to micrometer conversion. Record

this image to include in your lab write-up. Remove the sample and clean the surfaces before

continuing.

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Step 15. Fill one of the provided sample chambers with a bead solution. The TA will supply

a sample of 1 μm silica microspheres, which can be visualized using brightfield microscopy. The

sample chambers (made as shown in Figure 4) hold ~30 μL of solution. (Bead solution is in

phosphate buffered saline, pH 7.4 with 0.1% n-Dodecyl β-D-maltoside to inhibit aggregation.)

Step 16. Insert the slide and adjust the

focus to beads stuck to the surface.

Step 17. Turn on the trapping laser and

observe the beam pattern as you move

the focus. You should be able see a distinct

pattern when the trapping laser scatters

from each surface that it encounters. Move

through different focal positions and

understand when you are imaging inside the

glass coverslip, inside the sample chamber,

and inside the glass slide. If the pattern is not

symmetric, you may not be able to trap

anything! This is a good time to realign if

necessary. Record a few images showing

how this pattern changes the focal plane

moves through the glass surface to include

in your lab write-up.

Step 18. Adjust the trap and sample position to trap a 1 μm silica microsphere. Be patient

with this step. Find a bead that is diffusing (not stuck to the surface) and move the sample

chamber and/or the position of telescope lens 2 to trap the particle. Only beads that are

diffusing, nearly in focus and appear “dark” (slightly in front of the focal plane) can be

trapped. It may help to begin at higher laser power.

Figure 4. Sample chamber assemblv.

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Step 19. Acquire a set of 10s movies of trapped bead at different laser powers. For analysis

of these images, record the movies at the maximum time resolution and with a very small field of

view that you believe will always show the trapped bead. Note that at lower laser powers the trap

is weaker and the bead fluctuates at higher amplitude. This will be quantified to derive the trap

stiffness.

VI. Day Two – E coli. flagellar motor stall force measurement Step 1. Verify that the optical trap is functioning and that silica beads can be trapped.

Repeat any Day 1 steps as necessary.

Step 2. Prepare a sample chamber with motile E coli.

a) Load a ~15 μL aliquot of the grown culture into a flow chamber and allow it to incubate for ~5 min, upside-down, to permit bacterial attachment to the coverglass surface.

b) Flush the flow chamber with 300 μL of motility buffer (10 mM potassium phosphate, 0.1 mM EDTA, pH 7.4) using a Kimwipe for suction to remove dead cells in the sample and leave on the stuck, rotating bacteria.

Step 3. Load the sample chamber and use the brightfield image to focus on the surface

where you should see several bacteria rotating. If they are not motile, another sample

needs to be loaded.

Step 4. Record a short movie capturing the rotation, which you can later analyze to

determine the rotational radius, r (Figure 3B).

Step 5. Determine the stall force of the bacteria.

a) Set the laser current to ~375 mA (forming a relatively stiff trap) and move the sample holder (or the trap) such that the tip of the cell is trapped and rotation is halted.

b) Reduce the laser power until the cell escapes the trap and begins rotating. c) Record the trap stiffness at escape. d) If you cannot estimate the escape distance, assume that it is 100nm e) One can also start at low current and gradually increase until rotation is halted

Step 6. Repeat the measurement (Steps 4-5) on 10-20 bacteria to assemble adequate

statistics

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Optics and Materials Parts List Part Description Part Number* Tube lens f = 200 mm, Ø25mm, Achromat AC254-200-A Objective lens CFN Plan 100x NA=1.45 oil-immersion

objective (Out of production, but similar to Nikon CFI Plan Apo Lambda 100X Oil)

Condenser lens f = 50 mm, Ø25mm, Achromat AC254-050-A Field lens f = 150 mm, Ø25mm, Achromat AC254-150-A Collimating lens f = 20 mm, Ø25mm Aspheric ACL2520 White light LED 6500K color temperature, 800 mW MCWHL5 980 nm Laser Single mode fiber coupled laser diode,

200mW @ 980+/-5nm, in 14-pin butterfly package with built-in monitor photodiode, thermoelectric cooler, thermistor, and FC/APC fiber pigtail

QPhotonics QFBGLD-980-200

Laser collimation lens Fiber Collimation Package, 980nm, f = 11.16mm, FC/APC

F220APC-980

Laser Power supply Fiber-coupled laser diode controller including laser driver, temperature controller, 14-pin butterfly mount

QPhotonics QCombo-6305

Telescope Lens 1 f = 175 mm, Ø25.4mm, near IR anti-reflection coating

Newport KPX103AR.18

Telescope Lens 2 f = 150 mm, Ø50.8mm, anti-reflection coating 650-1050nm

AC508-150-B

Silver mirror Protected Silver Mirror, reflectance > 94% @ 450 nm - 20 µm

ME2-P01

Beamsplitter (hot mirror) Ø25.4mm, T>85%@850-645nm, R>90%@750-1200nm

FM01

Shortpass filter (hot mirror)

Ø25.4mm, T>85%@850-645nm, R>90%@750-1200nm

FM01

CCD Camera 2.8 megapixel, monochrome, 14 bit, fan-cooled charge coupled device

Photometrics Retiga-R3

Vertical grating sample Variable Line Grating Slide, 3" x 1" R1L3S6P Bead sample 1 µm silica microspheres, Bangs Labs SS04N E. coli RP437 Motile Eschericia coli strain RP437 a generous gift from the

Emonet lab f =focal length, Ø=diameter, λ=wavelength *ThorLabs parts numbers given unless otherwise specified

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980nm Laser Specifications Sheet

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VII. Lab-write up questions – Days 1 and 2 A) Please include all images specified in the Day 1 and Day 2 instructions.

B) Analyze your trapped bead movies using the Matlab code particle_track.m to obtain

the position variance and use these to derive the stiffnesses, 𝑘Iand 𝑘K using Equation 9.

Discuss any differences in stiffness in the two transverse directions.

C) (optional) Compare the stiffness calculated from the commented region at the bottom of

particle_track.m, which implements the power spectral density of the bead position

fluctuations. For more details, see Neuman, K.C., et al.††††† D) Transport Parameters

o Calculate the mass of a 1μm silica microsphere and use this to estimate the root

mean square velocity at 300K in the gas phase. Also estimate the root mean

square velocity of this particle at 300K in water. How far would you expect the

particle to travel in 1 ms?

o Use the Debye-Stokes-Einstein relation,𝐷 = TWXefgh

, where η is the solvent viscosity

and r is the radius of the particle, to estimate the diffusion coefficient for a 1μm

silica microsphere at 300K in water. How far would you expect the particle to travel

in 1 ms?

o Which of the above provides a better description for what you observed for

untrapped 1μm silica microspheres? Is one calculation wrong?

E) Use the optical power needed to stop the rotation of several bacteria, your force

calibration values converting laser power to stiffness, and the escape distance to find

an approximation to the torque exerted by each bacterial flagellum you measured.

Report the mean and standard deviation of this distribution.

VIII. Acknowledgements I would like to acknowledge Prof. Thierry Emonet for his generous gift of RP437 E Coli cells and

Jessica Johnston and Dr. Xiongfei Fu for help establishing the motility assay. I would also like to

thank Professor Matt Lang (Vanderbuilt Univ.) for original conception of this lab module experiment.

†††††“Optical Trapping,” K.C. Neuman and S. M. Block, Rev. Sci. Instrum. 75, 2787 (2004)