260 Questions

Brownian motion each image frame is less than the displacement fluctuations in the

ABEL trap due to its finite stiffness? (Assume the room temperature viscosity of water

is 0.001 Pa⋅s.)

6.7

Assume solutions of the form A(ω) exp(iωt) for the Langevin force at angular fre­

quency ω.

a

Derive an expression for the displacement x(t) at time t in an optical trap.

b

If the power spectral density G(ω)dω is defined as |A(ω)²|, derive an expression

for the mean squared displacement in terms of G.

c

Show that the power spectral density should be a Lorentzian function. (Assume

for “white noise” that G is a constant and that the equipartition theorem predicts

that the mean squared displacement at each separate frequency is associated with

a mean energy of kBT/​2.)

6.8

An experimental protocol was devised using BFP detection to monitor the lateral dis­

placement of a 200 nm diameter latex bead attached to a rotary molecular motor of

the bacterial flagellar motor (which enables bacteria to swim) via a stiff filament stub

to a live bacterium, which was free to rotate in a circle a short distance above the cell,

which itself is stuck firmly to a microscope coverslip. The motor is expected to rotate

at speeds of ~100 Hz and is made up of around ~20 individual subunits in a circle

that each are thought to generate torque independently to push the filament around.

a

What is the minimum sampling bandwidth of the QPD in order to see all of the

torque-​generating units?

In practice, it is difficult to make a completely stiff filament; in a separate experi­

ment using a completely unstiffened filament attached to a 500 nm diameter latex

bead, it was found that the filament compliance resulted in a relaxation drag delay to

bead movement following each ratchet of a few tenths of a microseconds, whereas

a 1000 nm diameter bead had an equivalent response time ~10 times slower.

b

Explain these observations and discuss which bead is the best choice to try to

monitor rotation mechanism of the flagellar motor.

It is possible to make some of the 20 ratchet subunits nonfunctional without

affecting the others.

c

How many subunits need to be made nonfunctional to detect individual activity

of each torque-​generating subunit?

d

New evidence suggests that there may be cooperativity between the subunits—​

how does this affect your previous answers?

6.9

AFM force spectroscopy and optical tweezers are both used to investigate single-​

molecule mechanics stretching single biomolecules, as well as observing domain

unfolding and refolding of modules inside the molecules. Explain with reasoning if

one technique is better.

6.10 What is a “molecular signature,” and why are they needed? The “sawtooth” pattern

of a force-​extension trace as obtained from AFM force spectroscopy on certain

molecules is an example of a molecular signature. Can you think of other molecular

signatures?

6.11 Single-​molecule force spectroscopy is normally performed on purified molecules

or on the surface of cells. Why? Under what circumstances experiments might be

performed inside living cells?

6.12 At a prestigious biophysics tools and techniques awards dinner, a helium balloon

escaped and got loosely trapped just under the ceiling. Assuming no lateral friction,

how long would it take a red laser pointer of 1 mW power output to push the balloon

10 m across the length of the dinner hall ceiling using forward photon pressure alone?

How would this change using a fancier 5 mW green laser pointer? Would it make sig­

nificant difference to encourage all the other ~500 people attending the awards dinner

to assist in getting out their laser pointers and performing this in parallel? (This is

what the author attempted in the not too distant past. It demonstrates the great merit

in doing theoretical calculations in advance of experiments.)

6.13 A membrane protein was imaged using AFM in contact mode. In one experiment, the

protein was purified and inserted into an artificial lipid bilayer on a flat surface. This