258 Questions

c

10 times the bead radius is 5.0 µm so, from the same trigonometry analysis as for

part (b), the end-​to-​end distance of the DNA is √(1.0² +​ 5.0²) –​ 0.5 =​ 4.6 µm. This is

4.6/​5.1 or ~90% of Rmax, so the binding of the NAP has shifted DNA toward rodlike

limit, that is, a stiff filament.

d

If we model the diffusion coefficient by the Stokes–​Einstein relation (see Equation

2.11) and crudely approximate the local diffusion of the DNA as that of a sphere

of radius lp, then the diffusion coefficient of the bead in this case is greater by an

order of magnitude. If we thus neglect the DNA drag entirely and assume the

drag will be mainly due to the bead itself, then its diffusion coefficient is:

~1.38 × 10^–​21 × 300 /​(6 × π × 0.001 × 0.5 × 10^–​6) =​ 4.4 × 10^–​11 m²/​s. The time

t taken to diffuse the 2D distance of 0.688 µm is from Equation 2.12 will be

given roughly by:

t =​ (0.688 × 10^-​6)²/​(4 × 4.4 × 10^–​11) =​2.7 × 10^–​3 s or ~3 ms. This is an order

of magnitude smaller than the sampling time. The answer to this question

is therefore maybe. Although the video-​rate imaging is not fast enough to

adequately track the bead from frame-​to-​frame, in a sense for this simple

analysis it does not matter since we can still measure what the maximum

bead displacement is by sampling enough frames, albeit relatively slowly

though in realtity the bead might move too quickly to be trackable per

se, so we would need to rely on the blur image to deduce the bead’s max­

imum extent. However, if we want to measure more nuanced properties like

dynamic affects, for example, how to monitor the real-​time changes to the

bead deflection as a NAP is added to the flow cell, then we would need to

definitively track the bead from frame-​to-​frame, so sampling at least an order

of magnitude faster.

6.8  SUMMARY POINTS

◾Optical tweezers, magnetic tweezers, and AFM can all probe single biomolecule

mechanics.

◾Magnetic tweezers and modified optical tweezers can probe single-​biomolecule

torque.

◾AFM imaging can generate sub-​nanometer precise topological information of bio­

logical samples.

◾Electrical forces can be used to monitor currents through natural ion channels in

cells and through artificial nanopores for biosensing.

◾Electric field can control the rotation and displacements of particles attached to

biological structures or of biomolecules directly.

◾Whole tissues can be mechanically studied using relatively simple stretch devices.

QUESTIONS

6.1

Ficoll is a synthetic polymer of sucrose used to change osmotic pressure and/​or viscosity

in biophysical experiments. A version of Ficoll with molecular weight 40 kDa had vis­

cosities relative to water at room temperature of [1, 5, 20, 60, 180, 600] corresponding

to (w/​v) % in water of [0, 10, 20, 30, 40, 50], respectively. If a fluorophore-​labeled anti­

body has a Stokes radius of 8 nm and the viscosity of water at room temperature is

0.001 Pa·s, estimate the molarity of the Ficoll needed to be present to observe labeled

antibodies in vitro unblurred in solution using a wide-​field epifluorescence micro­

scope of 1.45 NA capable of sampling at 40 ms per image frame.