Chaire d'Analyse Appliquée                    The DNA modelling COURSE

The specific problem of the Cyclization probability

Generally accepted that unstressed DNA can have quite large natural curvatures, especially when so called A-tracts (a particular base pair sequence) are present.

Issue is how such natural curvature effects cyclization. Three different unstressed shapes

  1. straight, with no intrinsic curvature,
  2. bent like a C,
  3. or with the shape of a S,
will require different amounts of energy to cyclize, and phasing of the end base-pairs obviously play a role in the process, as the double helix closure requires the individual strands to remain anti-parallel, which prevents the formation of a Moebius type of ribbon. 

Related questions arise in DNA-protein binding which is a basic biological mechanism that can produce sharp kinks in the DNA.

For example the 156 base pair molecule shown here has an approximately 110 degree bend in its minimum-energy stress-free uncyclized state.

The open configuration is generated as the minimum energy or unstressed state according to a modified Trifonov wedge angle model. Some such discrete information coming from molecular biology is the input to the continuum model.

And it cyclizes as shown:

 
 
 

The solid lines are the computed output of the continuum model, with the dots being the reconstructed wedge-angle equilibrium. Difference in energies between discrete and continuous models is $0.5\%$.

 
Biochemists can experimentally measure how the equilibrium constant between cyclized and uncyclized forms depends upon differences between unstressed shapes for various short DNA molecules.

(Actual measurement is ligated cyclization rates of cyclized vs dimer products)

According to various standard statistical mechanics formulæ the equilibrium constant is related to the difference in free energies between uncyclized and cyclized states.

The particular question we addressed is whether continuum rod computations of the internal (elastic) energy can correctly duplicate the experimentally measured differences for DNA molecules with differing unstressed shapes.

The first semester of the course will explain the static model used in these computations and the limits of the model applied here.
 
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