Chapter 6 – Forces  259

6.2

In a two-​bead optical tweezers tapping style mechanical stretch experiment on a single

molecule of linear DNA, 1600 separate bead pairs were generated over the course

of a week by a diligent student. They used a constant bead tapping frequency with

a triangle wave profile. The student thought there were three different populations

of molecules characterized by different estimated values of persistence length based

on worm-​like chain model fits applied to the force-​extension data (see Chapter 8),

which indicated 1497 molecules having a persistence length of ~50 nm. There are 39

molecules that had a persistence length of close to ~20–​30 nm, and the remainder

had a persistence length of ~10–​15 nm. For the group of molecules with persistence

length close to 50 nm, the DNA molecular stiffness was observed to decrease at values

of molecular force above ~65 pN. Explain these observations.

6.3

A “vertical” magnetic tweezers experiment was performed on a single molecule

of DNA tethered between a microscope coverslip and a magnetic bead, when the

molecular motor FtsK, which uses DNA as a track on which to translocate, was added

to the microscope sample chamber; the length of the distance between the coverslip

and the magnetic bead was observed to decrease.

a

Some researchers have used this as evidence that there might be two FtsK

molecular motors acting together on the DNA—​explain why this makes sense.

b

What other explanations could there be?

6.4

Optical tweezers using a focused laser of wavelength 1047 nm exerted a lateral force

of 80 pN on a latex bead of diameter 1000 nm suspended in water at room tempera­

ture when the bead is displaced 500 nm from the trap center.

a

If the trapping laser power passing through the bead is 220 mW, estimate the

average angle of deviation of laser photons, assuming the lateral force arises prin­

cipally from photons traveling close to the optical axis.

b

Make an annotated sketch of the frequency power spectral density of the

microsphere’s positional fluctuations.

c

At what frequency is the power spectral density half of its maximum value for

these optical tweezers? The incident laser beam is then divided up using a time-​

share approach with an AOD of efficiency 75% into several beams of equal power

to generate several independent optical tweezers.

d

If each optical trap must exert a continuous high force of 20 pN, estimate the

maximum number of traps that can be used.

e

For experiments not requiring continuous high force, estimate the maximum

theoretical number of optical traps that can be generated by this method if you

assume that a stable trap is such that mean displacement fluctuations of a trapped

particle position do not extend beyond the physical dimensions of the trap.

f

How many such trapped beads would be required to push on a single molecular

motor molecule to prevent it from undergoing a force-​generating molecular con­

formational change known as a power stroke of average magnitude 5 pN?

6.5

Most bacteria have an outer rigid cell wall composed of proteins and sugars (peptido­

glycan, see Chapter 2), which allows them to withstand osmotic pressures of 15 bar

or more, but semipermeable so allows a variety of small molecules, including water,

to diffuse through.

a

A virus known to infect bacteria has a diameter of 50 nm—​is it likely that the

reason why each bacterial cell ultimately splits after a virus infects the cell and

multiplies inside the cell is due to the buildup of pressure due to the large number

of virus particles?

b

If each virus consists of a maximum of 2000 protein molecules in its “capsid” coat

(see Chapter 2), which were to spontaneously split apart from each other, would

this make any difference to your answer? (Hint: treat a population of viruses in a

cell as an ideal gas whose concentration is limited by tight packing.)

6.6

An ABEL trap was used to constrain a 20 nm latex bead in water whose surface

contained the equivalent of ~3000 delocalized electrons. A mean E-​field strength of

9000 V m⁻¹ was applied to the trap’s electrodes. How many frames per second must

a camera sample the bead’s position to ensure that the expected distance diffused by