## Computer Simulations of Supercoiled DNA Energetics and Dynamics

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A new formulation is presented for investigating supercoiled DNA configurations
by deterministic techniques. Thus far, the computational difficulties involved
in applying deterministic methods to supercoiled DNA studies have generally
limited computer simulations to stochastic approaches. While stochastic
methods, such as simulated annealing and Metropolis-Monte Carlo sampling,
are successful at generating a large number of configurations and estimating
thermodynamic properties of topoisomer ensembles, deterministic methods
offer an accurate characterization of the minima and a systematic following
of their dynamics.

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To make this feasible, we model circular duplex DNA compactly by a B-spline
ribbon-like model in terms of a small number of control vertices. We associate
an elastic deformation energy composed of bending and twisting integrals
and represent intrachain contact by a 6-12 Lennard Jones potential. The
latter is parametrized to yield an energy minimum at the observed
DNA-helix diameter inclusive of a hydration shell. A penalty term to ensure
fixed contour length is also included. First and second partial derivatives
of the energy function have been derived by using various mathematical
simplifications. First derivatives are essential for Newton-type minimization
as well as molecular dynamics, and partial second-derivative information
can significantly accelerate minimization convergence through preconditioning.
Here we apply a new large-scale truncated-Newton algorithm for minimization
and a Langevin/implicit-Euler scheme for molecular dynamics. Our truncated-Newton
method exploits the separability of potential energy functions into terms
of differing complexity. It relies on a preconditioned conjugate gradient
method that is efficient for large-scale problems to solve *approximately*
for the search direction at every step. Our dynamics algorithm is numerically
stable over large timesteps. It also introduces a frequency-discriminating
mechanism so that vibrational modes with frequencies greater than a chosen
cutoff frequency are essentially frozen by the method.

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With these tools, we rapidly identify corresponding circular and interwound
energy minima for small DNA rings for a series of imposed linking-number
differences. These structures are consistent with available electron
microscopy data. The energetic exchange of stability between the circle
and the figure-8, in very good agreement with analytical results is also
detailed. Molecular dynamics trajectories at 100 femtosecond timesteps
then reveal the rapid folding of the unstable circular state into supercoiled
forms. Significant bending and twisting motions of the interwound structures
are also observed. Such information may be useful for understanding transition
states along the folding pathway and the role of enzymes that regulate
supercoiling. More generally, new quantitative data obtained by such deterministic
approaches may help in interpreting the effects of supercoiling on
key biological processes.

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