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DNA cyclization and looping in the wormlike limit: normal modes and the validity of the harmonic approximation

For much of the last three decades Monte Carlo-simulation methods have been the standard approach for accurately calculating the cyclization probability, $J$, or J factor, for DNA models having sequence-dependent bends or inhomogeneous bending flexibility. Within the last ten years, however, approaches based on harmonic analysis of semi-flexible polymer models have been introduced, which offer much greater computational efficiency than Monte Carlo techniques. These methods consider the ensemble of molecular conformations in terms of harmonic fluctuations about a well-defined elastic-energy minimum. However, the harmonic approximation is only applicable for small systems, because the accessible conformation space of larger systems is increasingly dominated by anharmonic contributions. In the case of computed values of the J factor, deviations of the harmonic approximation from the exact value of $J$ as a function of DNA length have not been characterized. Using a recent, numerically exact method that accounts for both anharmonic and harmonic contributions to $J$ for wormlike chains of arbitrary size, we report here the apparent error that results from neglecting anharmonic behavior. For wormlike chains having contour lengths less than four times the persistence length the error in $J$ arising from the harmonic approximation is generally small, amounting to free energies less than the thermal energy, $k_B T$. For larger systems, however, the deviations between harmonic and exact $J$ values increase approximately linearly with size.