Biological Physics: Energy, Information, Life

6.7. RNA folding as a two-state system[[Student version, January 17, 2003]] 201

14.0 pN

20 nm 1 s

13.6 pN

15.2 pN

14.6 pN

14.2 pN

14.1 pN

force, pN

fraction folded

13.5 14.0 14.5

0.2

0

0.4

0.6

0.8

1

extension, z,nm

force,

pN

5

10

15

20

(^100150200250)
f
folded
unfolded f
∆z=22 nm
P(f) =^1
1 + e−(∆F^0 −f∆z)/kBT
ab
a
c
b
d
dwell times for open state
closed state
dwell times for open state
closed state
14.4 pN 13.7 pN

10

30

50

70

0 1.0 2.0 3.0 4.0

10

30

50

0 1.0 2.0 3.0 4.0

10

20

30

dwell time, s

counts per bin

10

30

50

70

dwell time, s

Figure 6.10:(Experimental data) (a)Force-extension curves of an RNA hairpin with handles. Stretching(black)
and relaxing(gray)curves are superimposed. Hairpin unfolding occurs at about 14. 5 pN(labeled “a”). (b)Fraction
P(f)ofhairpins folded, versus force. Data(filled circles)are from 36 consecutive pulls of a single RNA hairpin.Solid
line,probability versus force for a two-state system (see Equation 6.34 on page 199). Best fit values, ∆F 0 =79kBTr,
∆z=22nm,consistent with the observed ∆zseen in panel (a). (c)Effect of mechanical force on the rate of RNA
folding. Length versus time traces of the RNA hairpin at various constant forces. Increasing the external force
increases the rate of unfolding and decreases the rate of folding. (d)Histograms of the dwell times in the open and
closed states of the RNA hairpin at two different forces (f=14. 4 and 13.7pN). The solid lines are exponential
functions fit to the data (see Equation 6.31), giving rate constants for folding and unfolding. At 13.7pNthe molecule
is mostly folded, withkopen=0. 9 s−^1 ,andkfold=8. 5 s−^1 .At14. 4 pNthe unfolded state predominates, with
kopen=7s−^1 andkfold=1. 5 s−^1 .[Figure kindly supplied by J. Liphardt.]

to see how the application of increasing force tilts the equilibrium of the system toward the longer,
unfolded form of the molecule. At forces slightly below the critical force, the molecule stayed mostly
in the short folded state except for brief excursions into the longer unfolded state (Figure 6.10c,
lower curves). When the force was held at 14. 1 pN,the molecule spent roughly equal times in either
state (≈ 1 s). Finally, at 14. 6 pN,the effect was reversed: The hairpin spent more time in the