  The elastic properties of semiflexible polymers are of great importance in
biology. There are experiments on biopolymers like double stranded DNA, which
twist and stretch single molecules to probe their elastic properties. It is
known that thermal fluctuations play an important role in determining molecular
elastic properties, but a full theoretical treatment of the problem of twist
elasticity of fluctuating ribbons using the simplest worm like chain model
(WLC) remains elusive. In this paper, we approach this problem by taking first
a mechanical approach and then incorporating thermal effects in a quadratic
approximation applying the Gelfand-Yaglom (GY) method for computing fluctuation
determinants. Our study interpolates between mechanics and statistical
mechanics in a controlled way and shows how profoundly thermal fluctuations
affect the elasticity of semiflexible polymers. The new results contained here
are: 1) a detailed study of the minimum energy configurations with explicit
expressions for their energy and writhe and plots of the extension versus Link
for these configurations. 2) a study of fluctuations around the local minima of
energy and approximate analytical formulae for the free energy of stretched
twisted polymers derived by the Gelfand Yaglom method. We use insights derived
from our mechanical approach to suggest calculational schemes that lead to an
improved treatment of thermal fluctuations. From the derived formulae,
predictions of the WLC model for molecular elasticity can be worked out for
comparison against numerical simulations and experiments.
