Positive supercoiling of DNA occurs when the right-handed, double-helical conformation of DNA is twisted even tighter (twisted in a right-handed fashion) until the helix begins to distort and "knot." Negative supercoiling, on the other hand, involves twisting against the helical conformation (twisting in a left-handed fashion), which preferentially underwinds and "straightens" the helix at low twisting stress, and knots the DNA into negative supercoils at high twisting stress.
Supercoiling is expected to occur in any process where the DNA helix must be topographically linearized (i.e. the helix must be uncoiled) for the function of a macromolecular assembly. Transcription is a prime example of such a situation, and topoisomerases are indeed influential in regulating transcriptional supercoiling. The figure below emphasizes the manner in which supercoils may be generated during transcription.
In part A, the top figure depicts the transcriptional machinery of the cell ("R") tracking along linear double-helical DNA. The "E" bars at the ends of the DNA represent either fixed points in the cellular DNA scaffold or the high viscosity of long DNA tails, both of which resist the natural relaxation of supercoiling. As the transcriptional assembly travels along the DNA, it must unwind the helix; otherwise the transcriptional machinery would need to travel around the DNA as it progressed, which would tangle transcript product around the DNA. The unwinding generates positive supercoils in front of the transcription site and negative supercoils behind it (bottom figure).
Figure B indicates the same generation of supercoils in the transcription of circular DNA, where "E" again represents either a fixed point or high viscosity.
Such supercoils are generated by both DNA transcription and DNA replication, requiring the relaxation of both positive and negative supercoils by different topoisomerases in order to prevent distortion of the DNA.
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