Force-Extension for DNA in a Nanoslit: Using an Effective Dimensionality to Map between the 3D and 2D Limits

The force-extension relation for a semi-flexible polymer such as DNA confined in a nanoslit is investigated and it is found that both the effective persistence length and the form of the force-extension relation change as the chain goes from 3D (very large slit heights) to 2D (very tight confinement). Generalizations of the Marko-Siggia relation appropriate for polymers in nanoconfinement are presented. The forms for both strong and weak confinement regimes are characterized by an \textit{effective dimensionality}. At low forces, the effective dimensionality is given by the correlations along the polymer in the plane of the confining walls. At high forces, the theoretical force must account for reduced conformation space. Together the interpolations give good agreement for all slit heights at all forces. As DNA and other semi-flexible biopolymers are commonly confined \textit{in situ} to various degrees, both the idea of an effective dimensionality and the associated generalized Marko-Siggia interpolations are useful for qualitatively understanding and quantitatively modeling polymers in nanoconfinement.