Application of Fourier Methods to Computer Simulation of Supercoiled DNA


     Fourier methods have been applied to model the overall geometry of a supercoiled DNA curve, the starting coordinates of which can be obtained from electron microscopy measurements or other theoretical simulations. Evenly spaced points on a known DNA curve are selected and subsequently used to model the original curve in terms of a finite Fourier series. Hence, a simple analytical curve expression, which closely resembles the initial data, is obtained for analysis and optimization.

    Energy minimization of DNA supercoils represented by such finite Fourier series has been performed using an elastic energy model. Minimized configurations are identified for interwound supercoils of 1000 bp at various values of the linking number difference, Lk. The configurational  profiles of writhing numbers (Wr) and energy vs. Lk are similar to those of optimized structures previously found with a B-spline representation of the DNA supercoil. However there are some notable differences that probably result from the more global nature of the Fourier modeling  compared to the B-spline. At the same Lk, a higher value of the writhing number and a lower energy are consistently obtained  with the finite Fourier series representation. Unlike the B-spline minimized structures and the configurations identified earlier by finite element analysis, only two families, the figure-eight and the loose interwound forms at low Lk, are found to overlap; there is no overlapping of interwound configurational families at higher Lk.

The optimized configurations of three-lobed 1000 bp branched DNA supercoils have also been identified. Different families of structures are found over a /\Lk range between -0.6 and 6.0. Family I, with three lobes of similar shape and size and with Wr0, occurs at low Lk values. Families II-VI, also with three lobes of similar shape and size, exist over a range of Lk from 1.0 to 4.6. The structures in families IIa-Va with one lobe larger than the others have similar Wr values at the same Lk but are slightly higher in energy. Families VII and VIII of minima found between Lk of 3.2 and 6.0 are characterized by one interwound and two open lobes, the later being similar to but smaller than the lobes in the simpler branched structures of families II-VI. The occurrence of branched interwound energy minima is consistent with the highly branched configurations of supercoiled DNA observed under the electron microscope.

Configurational snapshots of long partially relaxed DNA plasmids (6400 bp) taken from electron microscopic images are found to optimize to closely related linear and branched interwound forms with writhing numbers comparable to those of the nearly planar projected starting states. The energy minimization tends to straighten and maximize nonbonded contacts  between interwound double helical strands and to fix the size of the hairpin loops  emerging  from the interwound domains. The configurational similarities  of these minimized "random" starting states with the families of optimized configurations systematically identified at shorter chain lengths affirm the completeness of the search for energy minima. 

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