6.7. Excursion: “RNA folding as a two-state system” by J. Liphardt, I. Tinoco, Jr., and C. Bustamante
Though the system is in equilibrium, and so visits all of its available states, nevertheless many
systems have the property that they spend a long time in one class of states, then hop to the other
class and spend a long time there. In that case it makes sense to apply the definition of free energy
(Equation 6.32) to each of the classes separately. That is, we letFa,I=〈Ea〉I−TSa,I,where the
subscripts denote quantities referring only to classi.
Your Turn 6h
Adapt the result in the Example on page 198 to find that
PI
PII=e−∆F/kBT, (6.34)where ∆F≡Fa,I−Fa,II.Interpret your result in the special case where all the substates in each
class have roughly the same energy.Our result says that our simple formula for the population of a two-state system also applies to a
complex system, once we replace energy by free energy.^6
Just as in Section 6.6.2, we can rephrase our result on equilibration populations as a statement
about the rates to hop between the two classes of substates:
kI→II/kII→I=e∆F/kBT. complex 2-state system (6.35)6.7 Excursion: “RNA folding as a two-state system” by
J. Liphardt, I. Tinoco, Jr., and C. Bustamante
Recently, we set out to explore the mechanical properties of RNA, an important biopolymer. In
cells, RNA molecules store and transfer information, and catalyze biochemical reactions. We knew
that numerous biological processes like cell division and protein synthesis depend on the ability
of the cell to unfold RNA (as well as to unfold proteins and DNA), and that such unfolding
involves mechanical forces, which one might be able to reproduce using biophysical techniques. To
investigate how RNA might respond to mechanical forces, we needed to find a way to grab the ends
of individual molecules of RNA, and then to pull on them and watch them buckle, twist and unfold
under the effect of the applied external force.
Weused anoptical tweezerapparatus, which allows very small objects, like polystyrene beads
with a diameter of≈ 3 μm,tobemanipulated using light (Figure 6.9). Though the beads are
transparent, they do bend incoming light rays. This transfers some of the light’s momentum to
each bead, which accordingly experiences a force. A pair of opposed lasers, aimed at a common
focus, can thus be used to hold the beads in prescribed locations. Since the RNA is too small to
betrapped by itself, we attached it to molecular “handles” made of DNA, which were chemically
modified to stick to specially prepared polystyrene beads (Figure 6.9, inset). As sketched in the
inset, the RNA sequence we studied has the ability to fold back on itself, forming a “hairpin”
structure (see Figure 2.18 on page 46).
When we pulled on the RNA via the handles, we saw the force initially increase smoothly with
extension (Figure 6.10a, black curve), just as it did when we pulled on the handles alone: The DNA
(^6) T 2 Youcan generalize this discussion to the fixed-pressure case; the Gibbs free energy appears in place ofF
(see Section 6.6.3′on page 209).