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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!!!!