Chapter 6 – Forces  229

field strengths, the magnetization M (given by m/​V where V is the bead volume) saturates at

a value Mmax. For the most common permanent magnet pair arrangement, the B-​field is par­

allel to the focal plane of the microscope in between the opposite magnet poles, which means

that the B-​field gradient is zero everywhere apart from the vector normal to the focal plane.

Thus, the magnetic force is parallel to optic axis (z) in a direction away from the microscope

coverslip surface:

(6.15)

F

M

V

B

z

z ⁼

max

d

d

Thus, a biopolymer tethered between the coverslip and a magnetic bead will be stretched

vertically until balanced by the opposing molecular force that increases with molecular

extension.

The trapped bead’s position is still free to fluctuate in the lateral plane. Considering

displacements parallel to the focal plane, the small angle δθ satisfies

(6.16)

tanδθ =

=

x

z

F

F

x

z

where

z is the molecular extension of the biopolymer parallel to the optic axis

x is the displacement from the equilibrium in the focal plane

The equipartition theorem can be applied similarly as for optical tweezers to estimate the

stretching force parallel to the optic axis:

(6.17)

F

k T

x

z

z

= 〈

〉

B

2

Measurement of x can be achieved using similar techniques to optical tweezers bead detec­

tion, including bright-​field detection of the bead image onto a QPD or CCD, or to use BFP

detection that is less common for magnetic tweezers systems since it requires an additional

focused detection laser to be coaligned with the magnetic trap. As the bead moves above

and below the focal plane, its image on a CCD camera contains multiple diffraction rings.

The diameter of these rings is a metric for z, which can be determined by precalibration.

Measuring the torque on a tethered molecule requires knowledge not only of the magnetic

dipole moment and the local magnetic field strength but also the angle between their two

vectors. However, since a magnetic bead is spherically symmetrical, this can be difficult to

determine unless asymmetry is added, for example, in the form of a marker on the bead

for angle of rotation, such as a fluorescent quantum dot (see Chapter 3) fused to the mag­

netic bead.

The B-​field vector can be rotated either by differential phasing of the AC current input

through each different electromagnetic coil or by mechanically rotating the two permanent

magnets, which thus results in rotation of the magnetic bead. A paramagnetic bead may be

similarly rotated by first inducing a magnetic moment in the bead by the presence of a sep­

arate nearby permanent magnet.

Usually, the magnetic bead is conjugated to a single biomolecule of interest, which in

turn is tethered via its opposite end to a microscope slide or coverslip. By moving the stage

vertically relative to the permanent magnets or coils, for example, by changing the focus,

the molecule’s end-​to-​end extension can be controllably adjusted. Therefore, the mechanical

properties of individual molecules can be probed with this approach in much the same way

as for optical tweezers. One advantage of magnetic tweezers over optical tweezers is that

there is potentially less damage to the biological sample, since high stiffness optical tweezers

at least require a few hundred milliwatts of NIR laser power, which is sufficient to raise the

sample temperature and induce phototoxic effects.