The End-to-End Distance of RNA as a Randomly Self-Paired Polymer

1. The End-to-End Distance of RNA
as a Randomly Self-Paired Polymer
Li Tai Fang
Department of Chemistry & Biochemistry
UCLA

2. RNA
a biopolymer
consisting of 4
different species of
monomers (bases):
G, C, A, U
G–C
A –U secondary
G–U structure
3'
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

4. Association of 5' – 3' required for:
● Efficient replication ● Efficient translation
of viral RNA of mRNA
complementary RNA binding
sequence protein
e.g.,
HIV-1, Influenza, Sindbis, etc.

5. 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

6. Circle diagram

7. Circle diagram

8. Circle Diagram
● 60% of bases are paired
● duplex length ≈ 4
● Inspired the “randomly
self-paired polymer”
model

9. randomly self-paired polymer
e.g.,
NT = 1000 NT,eff = 550
Np = 600 Np,eff = 150

10. 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 – pj)·(1 – pk)· … ·(1 – pz)
(X) = P(X) – P(X–1) = probability R ee is X
X = X (X) · X

11. preview of the results:
Fang, L. T., J. Theor. Biol., 2011

12. Let's start the grunt work
Reminder:
RNA: Model:
NT = 1000 NT,eff = 550
Np = 600 Np,eff = 150
st
Now, the 1 challenge:

13. probability of a particular set of pairs
i j k l m n
p(i) = 150/550
p(i­j) = 1 /549 = p (this partial set)
p(k) = 148/548
= p(i) p(i – j) p(k) p(k – l) p(m) p(m – n)
p(k­l) = 1 /547
p(m) = 146/546
p(m­n) = 1 /545 depends on NT,eff, Np,eff, and B

14. Next challenge:
● We have pi = p(NT,eff, Np,eff, B)
● We want P(X) = 1 – (1 – pi)·(1 – pj)·(1 – pk)· … ·(1 – pz)
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
x1 x2 x3 x4
B = 3:
i j k l m n

15. Task: find (B)
● 1st, find the number of sets {x1, x2, …, xB+1},
such that X = x1+ x2+ … + xB+1
● for B = 3, X = 10: # of ways to arrange these:
X+B (X+B)!
=
B X!  B!

16. For each {xi}, how many ways to move the
middle regions?
vs.
i j k l i j k l
Navailable NT,eff – X – B – 1
=
B–1 B–1

17. Consider all X's
X
 X+B
B
NT,eff – X – B – 1
B–1
Xi=0
Missing something...... base-pairing “crossovers:”
(a) (b) (c) vs. (a) (b) (c)
i j k l i j k l

18. Crossovers are also known as pseudoknots
● X = xa + xb + 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

19. ( NT,eff , X, B )
X
= 2 (B – 1)
  X+B
B
NT,eff – X – B – 1
B–1
Xi=0

20. 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 – pj)·(1 – pk)· … ·(1 – pz)
P(X) = 1 – (1 – pB=1) B=1 · (1 – pB=2) B=2 · … · (1 – pBmax) Bmax
●
(X) = P(X) – P(X–1)

21. Probability distribution of end-to-end distances
<X> = 14.4
Fang, L. T., J. Theor. Biol., 2011

22. end-to-end distance vs. sequence length
X =  X (X) · X
Fang, L. T., J. Theor. Biol., 2011

23. 1/4
Scaling law: <X> ~ N
Fang, L. T., J. Theor. Biol., 2011

24. 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

25. Acknowledgment
● Thesis advisors
● Professors Bill Gelbart and Chuck Knobler
● Thanks to
● Professor Avinoam Ben-Shaul @ Hebrew University of
Jerusalem