A "relaxed," circular, double-stranded DNA molecule \((1600 \mathrm{bp})\) is in a solution where conditions favor 10 bp per turn. What is the value of \(L_{0}\) for this DNA molecule? Suppose DNA gyrase introduces 12 negative supercoils into this molecule. What are the values of \(L\) \(W,\) and \(T\) now? What is the superhelical density, \(\sigma\) ?

DNA topology refers to the spatial arrangement of a DNA molecule in three dimensions. It involves the twisting and coiling that occurs in DNA as it packs into the confined space within a cell. This organization is crucial because it can affect how DNA is read and copied during processes like replication and transcription.

In its simplest form, DNA exists as a double helix, where two strands weave around each other in a helical structure. The helix can undergo further twisting and coiling, leading to structures known as supercoils. Supercoiling occurs when the DNA helix strains against its natural coiling, either overwinding or underwinding, often as a result of enzymes that manipulate its structure or the processes of transcription and replication. The topology of DNA can impact its function - for example, certain topological forms can be more active transcriptionally, or more accessible to certain proteins or enzymes.

The linking number (L) is a fundamental concept in DNA topology that quantifies the level of twisting in the double helix. It represents the total number of times one strand of DNA winds around the other. The linking number is an invariant quantity for a closed circular DNA molecule unless the strand is physically cut. It can be determined by adding the number of twists (T), which is the number of turns in the helix, and the number of writhe (W), which is essentially the number of supercoils.

For relaxed DNA, the linking number equals the twist, because there is no supercoiling, hence W is zero. Any alteration in the number of supercoils, such as the introduction of negative or positive supercoils by enzymes, changes the linking number (L) and consequently the DNA's topology. In a mathematical form, this relationship is expressed as: \( L = T + W \).

Superhelical density (σ) is a ratio that describes the extent of supercoiling in DNA relative to a relaxed form of the molecule. It is calculated by dividing the writhe (W) by the linking number (L0) of the relaxed state. Technically, \( \sigma \text{ can be thought of as the } \frac{W}{L_0} \).

It offers insight into the tension of a DNA molecule; a negative σ indicates negative supercoiling, which means the DNA is underwound, while a positive σ signifies positive supercoiling, or an overwound state. For many biological processes, a certain level of supercoiling is necessary as it influences the accessibility and dynamics of DNA by affecting the binding of proteins such as transcription factors.

DNA gyrase is an essential enzyme in bacteria which introduces negative supercoils into DNA using energy from ATP hydrolysis. This activity falls under the broader category of topoisomerases, which change the supercoiling of DNA by making transient breaks in the DNA strands. Specifically, gyrase is a type II topoisomerase that can introduce negative superhelical turns even in the absence of external tension, unlike other topoisomerases that only relax overwound DNA.

By controlling the level of supercoiling, gyrase plays a critical role in managing the topology of bacterial DNA, making it compact enough for cell division, yet accessible for replication and transcription. Gyrase facilitates these processes by increasing the DNA's negative supercoils, thereby reducing the torsional strain and allowing the machinery of the cell to interact more easily with DNA strands.