Phase Transitions in Stretched Multi-Stranded Biomolecules

1
Phase Transitions in Stretched Multi-Stranded
Biomolecules

• Hemant Tailor
• Dept of Physics Astronomy (UCL)

2
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

3
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

4
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

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

5
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.

6
Transfer Integral Method (I)

• With the Transfer Matrices

7
Breakage Patterns

• Labels determine breakage pattern.

Intact
Frayed
Bubble
8
Transfer Integral Method (II)

• Transfer Matrices with the breakage pattern
  factor becomes
• Using the breakage patterns indices, and
  including the boundary conditions,

Almost ready to evaluate Z!!!!
9
Transfer Integral Method (III)

• Delta functions can now be represented as
• Eigenfunctions are defined by the following
  eigenvalue equations

Let the solving begin!!!!
10
Transfer Integral Method (III)
Here we have contracted most of the , the
contraction of will depend on the breakage
patterns
11
DNA Intact State

• All base-pairs are intact, so

12
DNA Intact State (II)
13
DNA Frayed State

• broken base-pairs where

14
DNA Frayed States (II)
15
DNA Frayed States (III)
16
DNA Bubble State

• intact base-pairs then broken base-pairs
  

17
DNA Bubble States (II)
18
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!!!!