Problem 13 - Biochemistry: Concepts and Connections

Question

A circular double-stranded DNA molecule contains 4200 base pairs. In solution, the molecule is in a B-form helix, with about \(10.5\) base pairs per turn. The DNA circle has 12 superhelical turns. What is its superhelix density \(\boldsymbol{\sigma}\) ?

Step-by-Step Solution

Answer

The superhelix density is 0.03.

Step 1: Calculate the Linking Number (Lk₀)

The linking number for relaxed DNA, Lk₀, is calculated by dividing the total number of base pairs by the number of base pairs per helical turn. Given the DNA has 4200 base pairs and 10.5 base pairs per turn:\[ \text{Lk}_0 = \frac{4200}{10.5} = 400 \]

Step 2: Determine the Total Linking Number (Lk)

The actual total linking number Lk for this DNA includes the superhelical turns. The superhelical turns add to the initial Lk₀. Given there are 12 superhelical turns, we calculate Lk as follows:\[ \text{Lk} = \text{Lk}_0 + \text{Superhelical Turns} = 400 + 12 = 412 \]

Step 3: Calculate Superhelix Density (σ)

The superhelix density \( \sigma \) is calculated by the formula:\[ \sigma = \frac{\text{Lk} - \text{Lk}_0}{\text{Lk}_0} \]Substituting the values calculated:\[ \sigma = \frac{412 - 400}{400} = \frac{12}{400} = 0.03 \]

Key Concepts

Linking Number

In the world of DNA structure, the linking number is a crucial concept. Think of it as the total number of times one strand of DNA winds around the other in a circular DNA molecule. In a perfectly relaxed DNA, this number tells us about its natural twisting tendency. It is denoted as "Lk" and calculated by considering how DNA naturally pairs into a helical shape.

For example, in the given DNA molecule with 4200 base pairs and assuming each helical turn consists of 10.5 base pairs (a common feature of B-form DNA), we can determine the linking number for relaxed DNA, represented as Lk₀, by dividing the total base pairs by the base pairs per turn.

• Lk₀ = 4200 ÷ 10.5 ≈ 400.

This calculation tells us about the DNA’s natural state before any additional twists or turns, known as superhelical turns, are employed. When additional twisting occurs, the linking number changes, reflecting how much the DNA is twisted beyond its relaxed state.

Superhelical Turns

Superhelical turns represent additional twists introduced into the DNA structure. These are important because they affect the overall tension and arrangement of the DNA molecule.

In this context, when we look at the superhelical turns, we're specifically referring to those twists existing beyond the natural structure defined by Lk₀.

• For the DNA in question, it is stated to have 12 superhelical turns.
• This means the DNA is not only wound according to its natural structure but has 12 extra twists.

The total linking number (Lk), which now includes these extra turns, is calculated by adding the number of superhelical turns to the relaxed linking number:

• Lk = Lk₀ + superhelical turns = 400 + 12 = 412.

Understanding this addition can help us comprehend how DNA can be compactly stored in cells and how it changes configurations during processes like replication.

Superhelix Density

The concept of superhelix density gives us insight into how supercoiled a DNA molecule is. It describes the intensity of supercoiling relative to the DNA's natural state.

Superhelix density, denoted as \( \sigma \), is given by:

• \( \sigma = \frac{\text{Lk} - \text{Lk}_0}{\text{Lk}_0} \)

In our specific example, with a total linking number (Lk) of 412 and a relaxed linking number (Lk₀) of 400, the superhelix density is calculated as follows:

• \( \sigma = \frac{412 - 400}{400} = \frac{12}{400} = 0.03 \)

This calculation gives a quantitative measure of the supercoiling present, suggesting that this DNA has slightly more than its natural coil configuration by 3%. Superhelix density is crucial because it affects the DNA's biological functions, including gene expression and DNA replication.