the end-to-end distance of rna as a randomly self-paired polymer

The End-to-End Distance of RNAas a Randomly Self-Paired Polymer

Li Tai Fang

Department of Chemistry & BiochemistryUCLA

RNA

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

GAG

secondarystructure

–––

CUU

5'3'

generic vs. sequence-specific properties

● Regardless of sequence or length, we can predict● Pairing fraction: 60%

● Average loop size: 8

● Average duplex length: 4

● 5' – 3' distance

Association of 5' – 3' required for:

● Efficient replication of viral RNA

● Efficient translation of mRNA

e.g., HIV-1, Influenza, Sindbis, etc.

complementary sequence

RNA bindingprotein

Question:How do the 5' and 3' ends of long RNAs find each other?Answer:The ends of RNA are always in close proximity, regardless of sequence or length !

Yoffe A. et al, 2010

Circle diagram

Circle Diagram

● 60% of bases are paired

● duplex length ≈ 4

● Inspired the “randomly self-paired polymer” model

randomly self-paired polymer

e.g., N

T = 1000

Np = 600

NT,eff

= 550

Np,eff

= 150

general approach

1) pi = probability that the ith set of “base-pair(s)”

-------will bring the ends to less than/equal to X

2) P(X) = at least one of those sets will occur

= 1 – (1 – pi)·(1 – p

j)·(1 – p

k)· … ·(1 – p

z)

(X) = P(X) – P(X–1) = probability Ree

is X

X = X (X) · X

preview of the results:

Fang, L. T., J. Theor. Biol., 2011

Let's start the grunt work

RNA:N

T = 1000

Np = 600

Model:N

T,eff = 550

Np,eff

= 150

Reminder:

Now, the 1st challenge:

probability of a particular set of pairs

i j k l m n

p(i) = 150/550p(ij) = 1 /549p(k) = 148/548p(kl) = 1 /547p(m) = 146/546p(mn) = 1 /545

= p (this partial set)

= p(i) p(i – j) p(k) p(k – l) p(m) p(m – n)

depends on NT,eff

, Np,eff

, and B

Next challenge:

● We have pi = p(N

T,eff, N

p,eff, B)

● We want P(X) = 1 – (1 – pi)·(1 – p

j)·(1 – p

k)· … ·(1 – p

z)

Let (B) = number of ways to make a set of pairs

Then, P(X) = 1 – (1 – pB=1

)B=1 · (1 – pB=2

)B=2 · … · (1 – pBmax

)Bmax

i j k l m n

B = 3: x

1 x

2 x

3 x

4

Task: find (B)

● 1st, find the number of sets {x1, x

2, …, x

B+1},

such that X = x1+ x

2+ … + x

B+1

● for B = 3, X = 10: # of ways to arrange these:

X + B ( X + B ) !

B X! B!=

For each {xi}, how many ways to move the

middle regions?

i j k l i j k l

vs.

Navailable

B – 1N

T,eff – X – B – 1

B – 1=

Consider all X's

X + B B

NT,eff

– X – B – 1

B – 1

X

X

i=0

Missing something...... base-pairing “crossovers:”

vs.

i j k l i j k l

(a) (b) (c) (a) (b) (c)

Crossovers are also known as pseudoknots

● X = xa + x

b + x

c

as long as xb j – i

____ and xb l – k

● 2 ways to connect each middle region

● undercount by 2(B – 1)

Now, let's put it all together

X + B B

NT,eff

– X – B – 1

B – 1

X

X

i=0

= 2(B – 1)

( NT,eff

, X, B )

Once again, the general approach

where end-to-end distance X

P(X) = at least one of these pairs will occur

P(X) = 1 – (1 – pi)·(1 – p

j)·(1 – p

k)· … ·(1 – p

z)

P(X) = 1 – (1 – pB=1

)B=1 · (1 – pB=2

)B=2 · … · (1 – pBmax

)Bmax

● (X) = P(X) – P(X–1)

Probability distribution of end-to-end distances


<X> = 14.4

end-to-end distance vs. sequence length

X = X (X) · X


Scaling law: <X> ~ N1/4


Once again:

● The ends of a self-paired polymer, such as RNA, are always in close proximity. ● This is a generic feature.

● Comparison of end-to-end distances:● random or worm-like polymers: X N1/2

● randomly branching polymers: X N1/4

● randomly self-paired polymers: X N1/8

Acknowledgment

● Thesis advisors

● Professors Bill Gelbart and Chuck Knobler

● Thanks to

● Professor Avinoam Ben-Shaul @ Hebrew University of Jerusalem

the end-to-end distance of rna as a randomly self-paired polymer

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