  We develop a statistical mechanical framework, based on a variational
approximation, to describe closed loop plectonemes. This framework incorporates
weak helix structure dependent forces into the determination of the free energy
and average structure of a plectoneme. Notably, due to their chiral nature,
helix structure dependent forces break the symmetry between left and right
handed supercoiling. The theoretical approach, presented here, also provides a
systematic way of enforcing the topological constraint of closed loop
supercoiling in the variational approximation. At large plectoneme lengths, by
considering correlation functions in an expansion in terms of the spatial mean
twist density about its thermally averaged value, it can be argued that
topological constraint may be approximated with by replacing twist and writhe
by their thermal averages. A Lagrange Multiplier, containing the sum of average
twist and writhe, can be added to the free energy to conveniently inforce this
result. The average writhe can be calculated through the thermal average of
Gauss integral in the variational approximation. Furthermore, this approach
allows for a possible way to calculate finite size corrections due to the
topological constraint. Using interaction energy terms from the mean-field
Kornyshev-Leikin theory, for parameter values that correspond to weak helix
dependent forces, we calculate the free energy, fluctuation magnitudes and mean
geometric parameters for the plectoneme. We see a slight asymmetry, where
interestingly enough left handed supercoils have looser structure than right
handed ones, although a lower free energy, unlike what previous ground state
calculations would suggest.
