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 =

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.

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.

Transfer Integral Method (I) With the Transfer Matrices: Need to apply breakage patterns and Boundary conditions!!! Not finished yet with !!!!!

Breakage Patterns Labels determine breakage pattern. 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, Almost ready to evaluate Z!!!!

Transfer Integral Method (III) Delta functions can now be represented as Eigenfunctions are defined by the following eigenvalue equations Let the solving begin!!!!

Transfer Integral Method (III) Here we have contracted most of the, the contraction of will depend on the breakage patterns

DNA Intact State All base-pairs are intact, so

DNA Intact State (II)

DNA Frayed State broken base-pairs where

DNA Frayed States (II)

DNA Frayed States (III)

DNA Bubble State intact base-pairs then broken base-pairs

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!!!!