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We develop a statistical mechanical framework, based on a variationalapproximation, to describe closed loop plectonemes. This framework incorporatesweak helix structure dependent forces into the determination of the free energyand average structure of a plectoneme. Notably, due to their chiral nature,helix structure dependent forces break the symmetry between left and righthanded supercoiling. The theoretical approach, presented here, also provides asystematic way of enforcing the topological constraint of closed loopsupercoiling in the variational approximation. At large plectoneme lengths, byconsidering correlation functions in an expansion in terms of the spatial meantwist density about its thermally averaged value, it can be argued thattopological constraint may be approximated with by replacing twist and writheby their thermal averages. A Lagrange Multiplier, containing the sum of averagetwist and writhe, can be added to the free energy to conveniently inforce thisresult. The average writhe can be calculated through the thermal average ofGauss integral in the variational approximation. Furthermore, this approachallows for a possible way to calculate finite size corrections due to thetopological constraint. Using interaction energy terms from the mean-fieldKornyshev-Leikin theory, for parameter values that correspond to weak helixdependent forces, we calculate the free energy, fluctuation magnitudes and meangeometric parameters for the plectoneme. We see a slight asymmetry, whereinterestingly enough left handed supercoils have looser structure than righthanded ones, although a lower free energy, unlike what previous ground statecalculations would suggest.

Statistical Mechanical Model for a closed loop plectoneme with weak helix specific forces