786
Zhang and Chen
3. Hall, K. B., and C. G. Tang. 1998. C-13 relaxation and dynamics of
the purine bases in the iron responsive element RNA hairpin.
Biochemistry. 37:9323–9332.
stack (see the shaded stack in Fig. 10 b), our rate constant
loop=kB
loop=kBT
model gives ka/b ¼ k0 eꢀDS
; kb/a ¼ k0 eꢀDH
;
stack=kB
and kb/c ¼ k0 eꢀDS
: The excess tetraloop stabilizatði0oÞn
4. Proctor, D. J., H. Ma, E. Kierzek, R. Kierzek, M. Gruebele, and P. C.
Bevilacqua. 2004. Folding thermodynamics and kinetics of YNMG
RNA Hairpins: specific incorporation of 8-bromoguanosine leads to
stabilization by enhancement of the folding rate. Biochemistry. 43:
14004–14014.
parameter can be determined as DSexcess ¼ DSloop – DSloop
loop
loop without the tetraloop stabilization.
and DHexcess ¼ DHloop, where DSð0Þ is the entropy of the
To directly connect the theory to the experiment, we con-
sider the YNMG RNA hairpins whose folding and unfolding
rates have been measured by Proctor et al. (4). We spe-
cifically compare the folding rates for the following two
sequences: ggacUUCGgucc (with tetraloop stabilization)
and ggacUUUUgucc (without tetraloop stabilization). To
extract the DSloop and DHloop for the experiment, we subtract
the stem parameters from the experimentally measured hairpin
parameters (4). Here the stem parameters are calculated from
the Turner rule (19) with the salt corrections (with experi-
mental condition of 10 mM Na1) (25).
5. Liphardt, J., B. Onoa, S. B. Smith, I. J. Tinoco, and C. Bustamante.
2001. Reversible unfolding of single RNA molecules by mechanical
force. Science. 292:733–737.
6. Zhang, W. B., and S. J. Chen. 2002. RNA hairpin folding kinetics.
Proc. Natl. Acad. Sci. USA. 99:1931–1936.
7. Zhang, W. B., and S. J. Chen. 2003. Master equation approach to
finding the rate-limiting steps in biopolymer folding. J. Chem. Phys.
118:3413–3420.
8. Sorin, E. J., M. A. Engelhardt, D. Herschlag, and V. S. Pande. 2002.
RNA simulations: probing hairpin unfolding and the dynamics of a
GNRA tetraloop. J. Mol. Biol. 317:493–506.
9. Sorin, E. J., Y. M. Rhee, B. J. Nakatani, and V. S. Pande. 2003.
Insights into nucleic acid conformational dynamics from massively
parallel stochastic simulations. Biophys. J. 85:790–803.
For the UUCG tetraloop, we found that DSexcess ¼ 25 eu
and DHexcess ¼ 12 kcal/mol. Proctor et al. (4) measured that
kfðexpÞ ¼ 6.1 3 104 sꢀ1 at T ¼ 65°C. Our theory (with Eq. 11)
gives kfðmodelÞ ¼ 8.91 3 104 sꢀ1, which is close to the ex-
10. Sorin, E. J., B. J. Nakatani, Y. M. Rhee, G. Jayachandran, V. Vishal,
and V. S. Pande. 2004. Does native state topology determine the RNA
folding mechanism? J. Mol. Biol. 337:789–797.
11. Cocco, S., J. F. Marko, and R. Monasson. 2003. Slow nucleic acid
unzipping kinetics from sequence-defined barriers. Eur. Phys. J. E. 10:
153–161.
perimental result. The unfolding rate can be estimated from the
hairpin stability DG(exp) ¼ – 0.79 kcal/mol as ku ’ kf eDG=k T
;
B
which gives kuðexpÞ ¼ 1.6 3 104 sꢀ1 and ku(model) ¼ 2.3 3
12. Bonnet, G., O. Krichevsky, and A. Libchaber. 1998. Kinetics of con-
formational fluctuations in DNA hairpin-loops. Proc. Natl. Acad. Sci.
USA. 95:8602–8606.
104 sꢀ1, respectively.
For the UUUU loop, there is no unusual tetraloop stabi-
lization interaction. By assuming DHexcess and DSexcess to be
13. Ansari, A., S. V. Kunznetsov, and Y. Shen. 2001. Configurational
diffusion down a folding funnel describes the dynamics of DNA
hairpins. Proc. Natl. Acad. Sci. USA. 98:7771–7776.
zero in the above equations (i.e., DHloop ¼ 0 and DSloop
¼
DSðlo0oÞp), we found that kfðmodelÞ ¼ 2:13 3ð1ex0p4Þ sꢀ1 at T ¼ 65°C,
14. Kuznetsov, S. V., Y. Shen, A. S. Benight, and A. Ansari. 2001. A
semiflexible polymer model applied to loop formation in DNA
hairpins. Biophys. J. 81:2864–2875.
which is close to the experimental result kf
¼ 4:5 3 104sꢀ1
:
’
The experimental and theoretical unfolding rates are kuðexpÞ
12:8 3 104sꢀ1 and kuðmodelÞ ’ 6:05 3 104sꢀ1; respectively.
15. Wallace, M. I., L. Ying, S. Balasubramanian, and D. Klenerman. 2001.
Non-Arrhenius kinetics for the loop closure of a DNA hairpin. Proc.
Natl. Acad. Sci. USA. 98:5584–5589.
Consistent with the experimental finding, the theory predicts
the acceleration in the folding process and the deceleration
in the unfolding process due to the tetraloop stabilization.
Physically, folding is accelerated because the excess intra-
loop stacking and basepairing can stabilize the transition
state for the folding (see z in Fig. 10 a) to lower the free
energy barrier of folding. The unfolding is decelerated be-
cause the intraloop stacking and basepairing in the folded
state can cause a higher (enthalpic) barrier for the disruption
of the tetraloop.
16. Wallace, M. I., L. Ying, S. Balasubramanian, and D. Klenerman. 2000.
FRET fluctuation spectroscopy: exploring the conformational dynam-
ics of DNA hairpin loop. J. Phys. Chem. B. 104:11551–11555.
17. Goddard, N. L., G. Bonnet, O. Krichevsky, and A. Libchaber. 2000.
Sequence-dependent rigidity of single-stranded DNA. Phys. Rev. Lett.
85:2400–2403.
18. Shen, Y., S. V. Kunznetsov, and A. Ansari. 2001. Loop dependence
of the dynamics of DNA hairpins. J. Phys. Chem. B. 105:12202–
12211.
19. Serra, M. J., and D. H. Turner. 1995. Predicting thermodynamic
properties of RNA. Methods Enzymol. 259:242–261.
We are grateful to Drs. Anjum Ansari and Herve Isambert for useful
discussions.
20. Chen, S. J., and K. A. Dill. 2000. RNA folding energy landscapes.
Proc. Natl. Acad. Sci. USA. 97:646–651.
This research was supported by the National Institutes of Health (NIH/
NIGMS) through grant GM No. 063732 (to S.-J. C).
21. Hilbers, C. W., C. A. Haasnoot, S. H. de Bruin, J. J. Joordens, G. A.
van der Marel, and J. H. van Boom. 1985. Hairpin formation in syn-
thetic oligonucleotide. Biochimie. 67:685–695.
22. Haasnoot, C. A., C. W. Hilbers, G. A. van der Marel, J. H. van Boom,
U. C. Singh, N. Pattabiraman, and P. A. Kollman. 1986. On loop
folding in nucleic acid hairpin-type structures. J. Biomol. Struct. Dyn.
3:843–857.
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