Exit Seminar:

Physics ofDNA, RNA, and RNA-like Polymers

Li Tai Fang

Department of Chemistry & BiochemistryUCLA

● DNA is a stiff,

self-repelling polymer● Capsid is highly

pressurized● DNA is released from

capsid upon binding LamB

Bacteriophage: DNA as genome

DNase

LamB

measure length:gel electrophoresis

Entropic Pulling Force

DNase

no pulling force:unaware of anoutside world

entropicpullingforce

Generic properties of DNA- independent of sequence

● Stiff and self-repelling● persistence length radius of capsid

● contour length diameter of capsid

● Confinement● entropy outside entropy inside

● Physical properties of DNA drive the initial infection process

RNA

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

GAG

secondarystructure

–––

CUU

5'3'

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

● Average loop size: 8

● Average duplex length: 4

generic vs. sequence-specific properties

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, 2009

Circle Diagram

Circle Diagram

Circle Diagram

● 60% of bases are paired

● duplex length ≈ 5

● Inspired the “randomly self-paired polymer” model

randomly self-paired polymer

e.g., N

T = 1000

Np = 600

NT,eff

= 520

Np,eff

= 120

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:

X

End-to-End Distances:

● flexible or worm-like polymers: X N1/2

– dsDNA, denatured RNA

● randomly self-paired polymers: X N1/4

– RNA-like polymer

● SuccessiveFold/MFold/Vienna: X N0

– RNA folding algorithms (no pseudoknot)

● flexible or worm-like polymers: X N1/2

– dsDNA, denatured RNA

● randomly self-paired polymers: X N1/4

– RNA-like polymer

● SuccessiveFold/MFold/Vienna: X N0

– RNA folding algorithms (no pseudoknot)

Let's start the grunt work

RNA:N

T = 1000

Np = 600

Model:N

T,eff = 520

Np,eff

= 120

Reminder:

Now, the 1st challenge:

probability of a particular set of pairs

i j k l m n

p(i)   = 120/520p(i­j) =  1 /519p(k)   = 118/518p(k­l) =  1 /517p(m)   = 116/516p(m­n) =  1 /515

= 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

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

bringing together the ends to Ree

X will

occur● P = none of those pair(s) actually occurs

= (1 – pi)·(1 – p

j)·(1 – p

k)· … ·(1 – p

z)

● Prob(X) = [1 – P(X)] – [1 – P(X–1)]

● X = X Prob(X) · X

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)

(X)

X

X = X (X) · X

X

Problems:

● Pseudoknots are rare in RNA● Not held in check in the self-paired

polymer model

● Successive RNA Folding Model:● Pseudoknots completely prohibited

Successive Folding Model:– created by Prof. Avi Ben-Shaul

randomly self-paired polymer Successive Fold

End-to-End Distances:– generic feature

● flexible or worm-like polymers: X N1/2

– dsDNA, denatured RNA

● randomly self-paired polymers: X N1/4

– RNA-like polymer

● SuccessiveFold/MFold/Vienna: X N0

– RNA folding algorithms (no pseudoknot)

Acknowledgment● Thesis advisors

Professors Bill Gelbart and Chuck Knobler

● Special thanks to

Professor Avi Ben-Shaul

● Thesis committee

Professors Joseph Loo, Giovanni Zocchi, Tom Chou

● Group members and former group members:

Aron Yoffe, Ajay Gopal, Odisse Azizgolshani, Peter Prinsen, Ruben Cadena, Cathy Jin, Maurico Comas-Garcia, Rees Garmann, Peter Stavros, Vivian Chiu Glover, Venus Vakhshori, Yufang Hu, Roya Zandi

B = 1:p = (120/520) (1/519) = 1/2249 = 4.45x10-4

= 231P(1) = (1 – 4.45x10-4)231 = 0.902

B = 2:p = (120 x 118) / (520x519x518x517) = 1.96x10-7

= 1.78 x 106

P(2) = (1 – 1.96x10-7)1.78E6 = 0.706

B = 3:p = 8.55x10-11

= 5.301x109

P(3) = (1 – 8.55x10-11)5.301E9 = 0.635

B = 4:p = 3.70x10-14

= 8.72x1012

P(4) = (1 – 3.70x10-14)8.72E12 = 0.725

For an RNA of N = 1000, pairing fraction = 0.6Probability that the ends will be no more than 20 unpaired bases apart?

Prob (X 20) = 1 – (0.902 0.706 0.635 0.725 … 1) = 0.81