Chapter 6 – Forces  219

This method was also employed to measure the mechanical properties of single DNA

molecules (Smith, 1996), which enabled estimation of the persistence length of DNA of

~50 nm based on worm-​like chain modeling (see Chapter 8) as well as enabling observations

of phenomena such as the overstretch transition in which the stiffness of DNA suddenly

drops at forces in the range 60–​70 pN due to structural changes to the DNA helix. Similarly,

optical tweezers have been used to measure the force dependence of folding and unfolding of

model structural motifs, such as the RNA hairpin (see Chapter 2, and Liphardt et al., 2001).

These techniques quantify the refolding of a molecule, indicating that they are far from a

simple reversal of the unfolding mechanism (see Sali et al., 1994).

Tethering a single biomolecule between two independent optically trapped beads

(Figure 6.3c) offers further advantages of fast feedback experiments to clamp both the

molecular force and position while monitoring the displacements of two separate beads

at the same time (Leake et al., 2004). Typically, a single-​molecule tether is formed by

tapping two optically trapped beads together, one chemically conjugated to one end of

the molecule, while the other is coated with chemical groups that will bind to the other

end. The two optically trapped beads are tapped together and then pulled apart over sev­

eral cycles at a frequency of a few hertz. There is, however, a probability that the number

of molecules tethered between the two beads is >1. If the probability of a given tether

forming is independent of the time, then this process can be modeled as a Poisson distri­

bution, such that probability Pteth(n) for forming n tethers is given by 〈〉

−〈〉

[

]

n n

n

n

exp

/ !,

with 〈〉

n the average number of observed tethers formed between two beads (see Worked

Case Example 6.1).

The measurement of the displacement of a trapped bead relative to the center of the optical

trap allows the axial force experienced by a tethered molecule to be determined from know­

ledge of the optical tweezer stiffness. The relationship between the force and the end-​to-​end

extension of the molecule can then be experimentally investigated. In general, the main con­

tribution to this force is entropic in origin, which can be modeled using a variety of polymer

physics formulations to determine parameters such as equivalent chain segment lengths in

the molecule, discussed in Chapter 8.

Several single-​molecule optical tweezers experiments are performed at relatively low forces

of just a few piconewtons, which is relevant to the physiological forces experienced in living

cells for a variety of different motor proteins (see Chapter 2). These studies famously have

included those of the muscle protein myosin interacting with actin (Finer et al., 1994), the

FIGURE 6.3  Tethering single biopolymers using optical tweezers. (a) A biopolymer, exem­

plified here by the giant molecule title found in muscle tissue, can be tethered between a

microscope coverslip surface and an optically trapped bead using specific antibodies (Ab1 and

Ab2). (b) A biopolymer tethered may also be formed between an optically trapped bead and

another bead secured by suction from a micropipette. (c) Two optically trapped beads can also

be used to generate a single-​molecule biopolymer tether, enabling precise mechanical stretch

experiments.