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

KIAS July 2006

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

KIAS July 2006

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

KIAS July 2006

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

KIAS July 2006

Simulation results

KIAS July 2006

N = 2 1024

KIAS July 2006

KIAS July 2006

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

KIAS July 2006

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

KIAS July 2006

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

KIAS July 2006

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

KIAS July 2006

Thank you!Thank you!

KIAS July 2006

KIAS July 2006

RNA World

floppy, ~1000 nt

Structural, recognition, catalytic, 150-4000 nt

adaptor, 75-95 nt

Various functions, e.g., RNA splicing

KIAS July 2006

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

KIAS July 2006

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.