Problem 4 - General Chemistry: Principles and Modern Applications

Question

Refer to a typical Escherichia coli bacterium. This is a cylindrical cell about \(2 \mu\) m long and \(1 \mu\)m in diameter, weighing about \(2 \times 10^{-12}\)g and containing about \(80 \%\) water by volume. A single chromosomal DNA molecule contains about 4.5 million nucleotide units. If this molecule were extended so that the nucleotide units were \(450 \mathrm{pm}\) apart, what would be the length of the molecule? How does this compare with the length of the cell itself? What does this result suggest about the shape of the DNA molecule?

Step-by-Step Solution

Answer

The DNA molecule is approximately 2.025 meters long, which is significantly longer than the bacterium in which it resides, suggesting it must be supercoiled within the cell.

Step 1: Calculating the length of the DNA molecule

The length of the DNA molecule can be calculated by multiplying the total number of nucleotide units, which is approximately 4.5 million, by the distance between these units, which is given as 450 pm, or picometers. To convert the distance between these units to a more accurate scale for this exercise, let's remember that 1 meter consists of \(10^{12}\) picometers. So, the complete formula will look like this: \(4.5 \times 10^{6}\) nucleotide units \(\times 450 \times 10^{-12}\) meters/nucleotide unit.

Step 2: Calculating and comparing

By performing the multiplications, it is found that the length of the DNA molecule is approximately \(2.025\) meters. Comparing this to the length of the bacterium cell, which is only \(2 \mu\)m, it becomes clear that the DNA molecule is much longer than the physical cell in which it resides. This could not be possible if the DNA molecule were stretched out straight within the cell.

Step 3: Interpreting the result

Given the fact that the DNA molecule is massively longer than the bacterium cell, it's indicative that the DNA molecule isn't stretched out in a straight line, rather it must be in a highly folded, or 'supercoiled' state, in order for such a long molecule to fit inside such a tiny cell.

Key Concepts

Nucleotide Units

DNA, the molecule that carries our genetic information, is composed of smaller structures known as nucleotide units. These units are the building blocks of DNA and play a crucial role in its structure and function.
Each nucleotide consists of three parts:

• The nitrogenous base, which can be one of four types: adenine (A), thymine (T), cytosine (C), or guanine (G).
• A sugar molecule, specifically deoxyribose.
• A phosphate group, which links the sugars to the bases to form a long chain.

The sequence of these nucleotide units encodes genetic information, which determines how organisms develop, function, and reproduce. Nucleotide sequences are read in groups of three called codons, each specifying a particular amino acid during protein synthesis. Understanding nucleotides is essential for grasping the broader picture of how genetic information is stored and transferred.

DNA Length Calculation

Determining the length of a DNA molecule involves understanding the number of nucleotide units it contains and the distance between these units. In the provided exercise about Escherichia coli, we learn it contains a DNA molecule with 4.5 million nucleotide units. These units are spaced 450 picometers (pm) apart.
To calculate the DNA length in meters, follow this process:

• Multiply the number of nucleotides (4.5 million) by the distance between nucleotides (450 pm).
• Convert picometers to meters, knowing 1 meter equals 1 trillion (\(10^{12}\)) picometers.
• Thus, the formula is: \(4.5 \times 10^{6} \times 450 \times 10^{-12} = 2.025\) meters.

This calculation reveals the DNA molecule is significantly longer than the cell itself, highlighting a fascinating aspect of molecular biology.

Supercoiling in Bacteria

Given the enormous length of DNA relative to the size of a bacterial cell, as calculated in earlier sections, it's clear that some way of compacting this genetic material is involved. Enter the concept of supercoiling.
Supercoiling refers to the overwinding or underwinding of a DNA strand, and is a part of the packing process that helps the DNA fit into the tiny space of a cell. In bacteria, like Escherichia coli, DNA does not exist in a stretched linear form. Instead, it is tightly coiled and looped.
The advantages of supercoiling include:

• Efficient packing of DNA to fit into a small cellular space.
• Protection from mechanical stress.
• Increased efficiency and regulation in gene expression and DNA replication.

This supercoiled state is crucial because it allows such lengthy DNA molecules to be stored within the compact environment of a bacterial cell, which only measures around 2 micrometers in length.