Sequential Folding ModelSequential Folding Modelpredicts length-independent secondary structure

properties of RNAs

Li Tai Fang

RNA

a biopolymer consisting of 4 different species of monomers (bases): G, C, A, U

GAG

secondarystructure

–––

CUU

Sequential Folding Model

Very simple Predicts generic secondary structure properties

of RNA Pairing fraction Duplex length Loop size Maximum ladder distance (MLD) → 3D size

Maximum Ladder Distance (MLD)

Approximating RNA into a linear polymer

The longest stretch determines its 3D size

MLD is a measure of 3D size: Rg ~ MLD1/2

Sequential Folding Model (SFM)

Created by A. Ben-Shaul

In each generation: find the longest duplex

Analytical solution

GAG

CUU

–––

Matching probability = 3/8

P(k) = (3/8)k

For k << NNumber of alignments of length k = (1/2)N2

On average, the longest duplex is determined by:(3/8)k1 (1/2)N2 = 1

k1 = a ln N – b, a = 2 / ln (8/3) ; b = ln 2 / ln (8/3)

CUU

k = duplex length

Average Duplex Length: Assumptions:

All divisions are symmetrical k << loop size in all generations (no longer valid

toward the end of the successive foldings)− Loop size in the sth generation of division: N

s = N / 2s

− ks = k

1 – (a ln 2)(s – 1)

s^ = the final generation k

s^ = 2, because 2 is the shortest stable duplex

ks^ + a ln 2

Pairing fraction: f

D = # of duplex; L = # of loops;

<k> = <duplex length>; <l> = <loop size>

D = L

D = N۰f / 2<k> ; L = N۰(1 – f) / <l>

f = 2 <k> / (<l> + 2<k>)

Assuming ks^ ~ 2 → <l> ~ 4

f ~ 0.68

Numeric results:predicted secondary structures

A) Vienna RNA B) SFM

Pairing fraction

Fang et. al., J. Phy. Chem. B (2011)

Duplex length

Fang et. al., J. Phy. Chem. B (2011)

Maximum ladder distance

Slope ~ 0.7

Fang et. al., J. Phy. Chem. B (2011)

Current Work: Kramer's theorem: mapping RNA secondary structure into

an “ideal” branched polymer

Acknowledgement:

– Avi Ben-Shaul: conceptualized the model

– Bill Gelbart

– Aron Yoffe