Ground state and the glass transition of the RNA secondary structure
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KIAS July 2006
Ground state and the glass transition of the
RNA secondary structureRNA secondary structure
• RNA folding: specific versus nonspecific pairing
• Ground state and finite temperature properties
• Logarithmic energy scale
• Distribution of pairing distances
• Summary
Tony Hui and Lei-Han Tang
Department of Physics, Hong Kong Baptist University
KIAS July 2006
Increasing complexity
and designability
Complementary
Partially complementary
sequence specific
Conformational Characteristics of Biopolymers
Can equilibrium statistical mechanics be of help in understanding bio-specificity?
KIAS July 2006
RNA Secondary Structures
R. Bundschuh and U. Gerland
Eur. Phys. J E 19, 319 (2006)
Iterative computation of partition function for a finite chain
, 1 /
, 1 , , 1 1,k j
jT
i j i j i k k jk i
Z Z Z e Z
RNA: single strand molecule of four different nucleotides.
Secondary structure: self-matching of the bases.
N3 algorithmpairing energy
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The phase diagramBundschuh and Hwa, PRL 83, 1479 (1999); PRE 65, 031903 (2002).
Tg
Low T: sequence specific pairing
High T: nonspecific pairing
for base
pairing
Michael Laässig and Kay Jörg Wiese, PRL 96, 228101 (2006)
gT T
gT T
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Analogy with the directed polymer problem
Transfer matrix power-law algorithm
Ground state scaling properties
But what are the values of the exponents for RNA?
Finite temperature transition
Role of disorder distribution
E L
x L
x
L
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Pairing energy
(a) Allowing only Watson and Crick pairing A-U and G-C, but no cooperativity
Extensive g.s. entropy for a typical random sequence
(Higgs, PRL 76, 704 (1996); Pagnani et al. PRL 84, 2026 (2000).)
(b) More realistic energy model (as in Zuker’s Mfold) with stacking energies etc.
force pairing to be at least several nucleotides long (a stem), matching of “words”
(c) Effective model: after coarse graining, we may assume to be independently distributed. A convenient distribution is
ij
0/10 , 0;
( )0, 0.
eP
Adequate for random
sequences
KIAS July 2006
Pinching (free) energyBundschuh and Hwa, PRL 83, 1479 (1999); PRE 65, 031903 (2002).
( ) ( )2
A BN N N NF F F F
(a) Random fluctuations of bond energies largely cancel out probing the effect of a perturbation on large scale.
(b) Above the glass transition,
(c) A different behavior is expected below
3ln
2NF T N const
gT
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Simulation results
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N = 2 1024
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Finite temperatures
2ln( / ) ( ) ln ( ) lnA B A BF kT Z Z Z A T N B T N
20( ) ,
( )0,
g g
g
A T T T TA T
T T
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One or two energy scales on each length scale?
Suppose the energy cost due to finite size is proportional to ln N. On each scale, only one such cost is warranted. To insert a break in the middle of the chain, bases close to the mid-point are affected. Hence the energy cost is equal to the sum of costs upto scale N, i.e.,
Pairing of bases at the end of a sequence is
limited
20 01
ln lnN
N
dnE A n A N
n
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Why logarithm?
N
Minimum of N realizations of the pairing energies , 1k N
min
/ 0
min
Prob( )
Prob( ) Prob( )
1-Prob( )
x
N
N
N x
Ne
x x
x
e
e
min 0Hence, ( ) lnN N (energy gain)
When 1, ,k N pairing with base N+1 splits the chain into two parts.
For the pairing to be favorable, we need min ( ) 0kE k
If , then 1.kE k k
But this implies pairing will always occur at short distances, in which case the power-law growth is false.
KIAS July 2006
Power-law pairing energies
110 0/ , 0;
( )0, 0.
P
Minimum of N2 realizations of the pairing energies ij
min
2
2
2
20
min
Prob( )
/
Prob( ) Prob( )
1-Prob( )
N
N
N x
N x
x x
x
e
e
2 /min 0Hence, ( )N N
2,3,4
Mean and width scale in the same way
KIAS July 2006
The surprise
Scaled distribution of the site who pairs with the end site
exponential tail
Power-law tail at =2
4 / 3( )P d d
Distribution of pairing distance
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Summary
• RNA secondary structure an interesting topic in statistical mechanics, with properties similar to the directed polymer problem. It has a low temperature phase with sequence specific pairing.
• Ground state energy of a finite chain contains a log-squared finite size term.
• The log-square term persists up to the glass transition, with its coefficient vanishing as the square of the distance to the transition.
• Since the energy cost for “remodeling” the pairing pattern grows logarithmically with chain length, two inserted sequences with particularly good matching can easily pair each other, at least under equilibrium conditions. This observation may be of some importance for rRNA’s.
• Distribution of the pairing distance assumes a power law with an exponent 4/3, quite independent of the pairing energy distribution.
• Analytical treatments? (cf recent attempt by M. Lassig and K. Wiese.)
ij
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Thank you!Thank you!
KIAS July 2006
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RNA World
floppy, ~1000 nt
Structural, recognition, catalytic, 150-4000 nt
adaptor, 75-95 nt
Various functions, e.g., RNA splicing
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The RNA folding problemprimary
3D structure with tertiary
contacts
GCGGAUUUAGCUCAGDDGGGAGAGCGCCAGACUGAAYACUGGAGGUCUGUGT CGAUCCACAGAAUUCGCACCA
Information flow (from sequence to structure) is hierarchical and sequential.
How RNA folds? Tinoco and Bustamante, JMB 293, 271 (1999)
secondary
base pairing
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The random energy modelB. Derrida, Phys. Rev. B 24, 2613 (1981)
min
min
/ / / ( )iE T E T E T
Ei
Z e e e E dE
glass transition: switching of the dominant term at
/ 2lngT N
2
2min
: ln / 2 ln
: /g
g g
T T F T Z T T N
T T F E T
annealed average
N energy levels drawn independently from a probability distribution function (E)
E
Emin
rare
typical
Thermodynamic limit: 0 lnN
( ) lnF A T N
20
0
0
, 22( )
2
g
g
T T TTA T
T T
Engineer the DOS
KIAS July 2006
assigning thermodynamic parameters to basic secondary structural components
(>1000!)
# stack_energies
/* CG GC GU UG AU UA
-200 -290 -190 -120 -170 -180
-290 -340 -210 -140 -210 -230
-190 -210 150 -40 -100 -110
-120 -140 -40 -20 -50 -80
-170 -210 -100 -50 -90 -90
-180 -230 -110 -80 -90 -110
Zuker’s mfold
Task:
Identify the ground state configuration among all possible pairing patterns.
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