24. A closed circular B-DNA of 4000 base pairs is negatively supercoiled by introduction of 4
writhes. The super helical density of the resultant DNA molecule will be ____

Understanding DNA Superhelical Density in a 4000 bp Negatively Supercoiled B-DNA Molecule

Closed circular B-DNA with 4000 base pairs becomes negatively supercoiled when 4 writhes are introduced, altering its topology. The superhelical density σ measures this supercoiling quantitatively. For this molecule, σ equals -0.01, reflecting mild underwinding typical in biological contexts.​

Superhelical Density Formula

Superhelical density σ is defined as σ = Wr / Tw₀, where Wr is writhe (superhelical turns) and Tw₀ is the relaxed twist number. B-DNA has 10.5 base pairs per helical turn, so Tw₀ = total bp / 10.5. Here, Tw₀ = 4000 / 10.5 ≈ 381 turns, and Wr = -4 (negative for underwinding), yielding σ = -4 / 381 ≈ -0.0105, or -0.01.​

Step-by-Step Calculation

First, compute Tw₀: 4000 bp ÷ 10.5 bp/turn = 380.95 turns. Negative supercoiling adds Wr = -4, so linking difference ΔLk ≈ Wr (assuming minimal twist change). Thus, σ = -4 / 380.95 ≈ -0.0105, rounded to -0.01 for standard reporting. This value indicates 1% underwinding, common in cellular DNA (σ ≈ -0.05 to -0.07 naturally).​

Biological Relevance

Negative supercoiling facilitates processes like transcription and replication by reducing energy barriers for strand separation. Enzymes like DNA gyrase introduce such writhes in vivo, maintaining σ around -0.06. The given -0.01 represents constrained supercoiling from 4 writhes.​

Common Exam Options Explained

• -0.01 (Correct): Matches σ = -4 / (4000/10.5) precisely.​

• -0.05: Typical cellular value, but exceeds 4 writhes (would need ~19).​

• -0.06: Natural bacterial average, requiring ~23 writhes.​

• 0 (Relaxed): No writhes introduced.​

• +0.01: Positive supercoiling (overwinding), opposite of negative.​