218 6.3  Optical Force Tools

The power spectral density emerges from the Fourier transform solution to the bead’s

equation of motion (Equation 6.7) in the presence of the random, stochastic Langevin force.

Here, ν0 is the corner frequency given by k/​2πγ. The corner frequency is usually ~1 kHz, and

so provided the x position is sampled at a frequency, which is an order of magnitude or more

greater than the corner frequency, that is, >10 kHz, a reasonable fit of P to the experimental

power spectral data can be obtained (Figure 6.2d), allowing the stiffness to be estimated.

Alternatively, one can use the equipartition theorem of thermal physics, such that the mean

squared displacement of an optically trapped bead’s motion should satisfy

(6.11)

k x

k T

k

k T

x

〈

〉=

∴

= 〈

〉

2

2

2

2

B

B

Therefore, by estimating the mean squared displacement of the trapped bead, the trap

stiffness may be estimated (Figure 6.2e). The Lorentzian method has an advantage in that it

does not require a specific knowledge of a bead displacement calibration factor for the pos­

itional detector used for the bead, simply a reasonable assumption that the response of the

detector is linear with small bead displacements.

Both methods only generate estimates for the trap stiffness at the center of the optical trap.

For low-​force applications, this is acceptable since the trap stiffness is constant. However,

some single-​molecule stretch experiments require access to relatively high forces of >100 pN,

requiring the bead to be close to the physical edge of the trap, and in this regime there can

be significant deviations from a linear dependence of trapping force with displacement. To

characterize, the position of an optical trap can be oscillated using a square wave at ~100 Hz

of amplitude ~1 μm; the effect at each square wave alternation is to rapidly (depending on the

signal generator, in <10⁻⁷ s) displace the trap focus such that the bead is then at the very edge

of the trap almost instantaneously. Then, the speed v of movement of the bead back toward

the trap center can be used to calculate the drag force; using Equation 6.7 and averaging

over many cycles such that the mean of the Langevin force is zero imply that the average

drag force should equal the trap restoring force at each different value of x, and therefore

the trap stiffness can be characterized for the full lateral extent of the trap. Similarly, optical

tweezers can be scanned across a surface-​immobilized bead in order to determine the pre­

cise response of the BFP detector at different relative separations between a bead center and

optical trap center.

6.3.4  APPLICATIONS OF OPTICAL TWEEZERS

Appropriate latex or silica-​based microspheres suitable for optical trapping can be commer­

cially engineered to include a chemical coating of a variety of different compounds, most

importantly carboxyl, amino, and aldehyde groups that can be used as adapter molecules

to conjugate to biomolecules. Using standard bulk conjugation chemistry, these chemical

groups on the bead surface can be bound either directly to biomolecules or more com­

monly linked to an adapter molecule such as a specific antibody or a biotin group that

will then bind to a specific region of a biomolecule of interest (see Chapter 7). Chemically

functionalizing microspheres in this way allows single biomolecules to be attached to the

surface of an optically trapped bead and tethered to a fixed surface such as a microscope

coverslip (Figure 6.3a).

Several of the first optical tweezers experiments involved the large muscle protein titin

(Tskhovrebova et al., 1997), which enabled the mechanical elasticity of single titin molecules

to be probed as a function of its molecular extension by laterally displacing the microscope

stage to stretch the molecule relative to the trapped bead. This technique was further modi­

fied to tether a single titin molecule between an optically trapped bead and a micropipette,

which secured to a second bead attached to the other end of the molecule by suction forces

(Figure 6.3b), which offered some improvement in fixing the tether axis to be parallel to the

lateral plane of movement of the trap thus making the most out of the lateral trapping force

available (Kellermayer et al., 1997).