Phase Transitions in Stretched Multi-Stranded Biomolecules

Hemant Tailor

Dept of Physics & Astronomy (UCL)

Introduction

• Use statistical field theory to study the nucleation of breakage of DNA under strain by external forces.

• More specifically, we are looking at a shearing problem where the opposite ends of the two back-bone strands of the DNA are pulled apart along its axis.

Biologically, RNA synthesis is an example of

where external forces act on DNA.

DNA-related nanotechnology – using DNA as

Nano-structured devices

DNA Toy Model – Geometric Representation• Represent DNA as a 1-D ladder structure

• Interactions along the backbone and base-pair are assumed to be harmonic.

• Each base-pair interacts through a potential which is dependent on the axial pair separation .

• Backbone spring constant = , Base-pair spring constant =

V

ii yx

01 y 2y 3y 4y 5y Ny

1x 2x 3x 4x 5x uxN

DNA Toy Model - Hamiltonian• The substitution of and were used to decouple

the variables for the integration.• The following transformations were also applied to simplify the

calculation:

• The Hamiltonian becomes:

N

ii

N

iii

N

iii VH

1

1

1

21

1

1

21 )()()(

• Inserting this into the configuration Integral we get an expression for Z that can be evaluated for different breakage patterns.

N

iii

H ddeZ ii

1

),(

iii yx iii yx

21

4

kT 2

1

4

kT

DNA Toy Model - Potential• The potential for the base-pair interactions is approximated by a

harmonic potential between the cut-offs .

• Outside the cut-off we consider the bond broken, and hence the potential is constant.

bb ,

2

2

1)( V

2

2

1)( bbV

)(V

b b

Transfer Integral Method (I)

2/)()(exp

),(ˆ),(ˆ),(ˆ),(ˆ

),(),(),(),(

1

14332211

14332211

1

),(

N

NN

N

ii

NN

N

ii

N

iii

H

VV

TTTTd

TTTTd

ddeZ ii

• With the Transfer Matrices:

2exp),( T

2/)()(expexp),(ˆ 2 VVT

Need to apply breakage patterns and Boundary conditions!!!Not finished yet with !!!!!),(ˆ T

Breakage Patterns• Labels determine breakage pattern. i

0,0,0,0,0i

0,0,0,1,1i

0,1,1,0,0i

Intact

Frayed

Bubble

Transfer Integral Method (II)• Transfer Matrices with the breakage pattern factor becomes:

• Using the breakage patterns indices, and including the boundary conditions,

2exp),( T

2/)()(expexp)()(),(ˆ 2, VVggT

)2()( 2/)()(exp

),(ˆ),(ˆ),(ˆ),(ˆ

),(),(),(),(

111

1,43,32,21,1

14332211

1433221

321

uVV

TTTTd

TTTTdZ

NNN

NN

N

ii

NN

N

ii

NN

N

)2()()()( 111 uuxy NNN

Almost ready to evaluate Z!!!!

g

Transfer Integral Method (III)• Delta functions can now be represented as

• Eigenfunctions are defined by the following eigenvalue equations

)(~~)(~),(ˆ

)(ˆˆ)(ˆ),(ˆ

)()(),(

11

00

vvv

ttt

sss

Td

Td

Td

s

ss )()()( 11*

11

t

NtNtNN uu )(ˆ)2(ˆ)2( *

Let the solving begin!!!!

Transfer Integral Method (III)

tNtNt

sss

N

NN

N

ii

NN

N

ii

u

VV

TTTTd

TTTTdZ

NN

N

)(ˆ)2(ˆ)()(

2/)()(exp

),(ˆ),(ˆ),(ˆ),(ˆ

),(),(),(),(

*11

*

1

1,43,32,21,1

14332211

1433221

321

2/)2()(exp

),(ˆ),(ˆ),(ˆ),(ˆ)(ˆ)(

)2(ˆ)(

1

1,43,32,21,1

1

**

,

1

1433221

321

N

NNNt

N

isi

NtNsNts

Ns

uVV

TTTTd

udZ

NN

N

Here we have contracted most of the , the contraction of will depend on the breakage patterns

i i

)()(),( sssTd

DNA Intact State

2/)2()(exp

),(ˆ),(ˆ),(ˆ),(ˆ)(ˆ)(

)2(ˆ)(

1

1004300320021001

1

**

,

10,...0,0,0

N

NNNt

N

isi

NtNsNts

Ns

uVV

TTTTd

udZ

)(ˆˆ)(ˆ),(0̂0 tttTd

• All base-pairs are intact, so 0i

)(ˆ 2/)(exp)(

)2(ˆ2/)2(exp)(ˆ

1111

**

,

110,...0,0,0

ts

NtNNsNts

Nt

Ns

Vd

uuVdZ

DNA Intact State (II)

DNA Frayed State

2/)2()(exp

),(ˆ),(ˆ),(ˆ),(ˆ),(ˆ)(ˆ)(

)2(ˆ)(

1

100540011011121111

1

**

,

10,...0,0,1,1

N

NNnnnnNt

N

isi

NtNsNts

Ns

uVV

TTTTTd

udZ

• broken base-pairs where 1i

)(ˆ),(ˆ)(~

)2(ˆ2/)2(exp)(~ˆ

11101

**

,

1110,...0,0,1,1

tvnn

NtNNsNts

sv

nv

nNt

Ns

Tdd

uuVdcZ

)(~)2/)(exp()( 111 vs

svs cV

)(~~)(~),(1̂1 vvvTd

n

DNA Frayed States (II)

DNA Frayed States (III)

DNA Bubble State

2/)2()(exp),(ˆ),(ˆ

),(ˆ)...,(ˆ),(ˆ),(ˆ)(ˆ)(

)2(ˆ)(

1100110

11121111011001

1

**

,

10,...1,1,0,0

NNNnlnl

nlnlllllllNt

N

isi

NtNsNts

Ns

uVVTT

TTTTd

udZ

• intact base-pairs then broken base-pairs

)(ˆ),(ˆ)(~

)(~),(ˆ)(ˆ

)2(ˆ2/)2(exp)(~ˆˆ

11101

*01'

**

,',,'

11'

110,...1,1,0,0

ltlllvnlnl

zvzlltzl

NtNNsNvtts

st

nv

lt

nlNt

Ns

Tdd

yyTdyd

uuVddZ

l n

DNA Bubble States (II)

Conclusion• Still work in progress!!!!• Successfully calculated Free Energies for Intact, Frayed and

Bubble states as a function of strand extension• See Phase Transitions from the free energy graphs• Next is to apply a similar Toy model for Collagen (Triple Helix) -

“Toblerone” as our geometric model!!!!