Problem 6 - Biochemistry: Concepts and Connections

Question

The average human chromosome contains about \(1 \times 10^{8}\) bp of DNA. (a) If each base pair has a mass of about 660 daltons and there are about \(2 \mathrm{~g}\) of protein (histones plus nonhistones) per gram of DNA, how much does such a chromosome weigh (in grams)? (b) If the DNA were extended, how long would it be? (c) An actual chromosome is about \(5 \mathrm{~mm}\) long. What is the approximate compaction ratio? (d) You have about \(4 \times 10^{12}\) cells in your body. If you have 46 chromosomes in each cell, what is the approximate extended length of all of your DNA? For comparison, the distance from the earth to the sun is about \(1.5 \times 10^{8} \mathrm{~km}\).

Step-by-Step Solution

Answer

(a) Chromosome weight is \(3.3 \times 10^{-13}\) grams; (b) Extended DNA length is 3.4 cm; (c) Compaction ratio is approximately 6.8; (d) Total DNA length is about 6.256 billion km, much longer than Earth-Sun distance.

Step 1: Understanding Base Pair Weight

The mass of one base pair is given as 660 daltons. Since 1 dalton is approximately \(1.66 \times 10^{-24}\) grams, the mass of one base pair in grams is \(660 \times 1.66 \times 10^{-24}\).

Step 2: Calculating Chromosome Mass

A chromosome contains \(1 \times 10^8\) base pairs. Multiplying the number of base pairs by the mass of one base pair gives us the total DNA mass in the chromosome: \((1 \times 10^8) \times (660 \times 1.66 \times 10^{-24})\) grams. This results in approximately \(1.1 \times 10^{-13}\) grams of DNA. Including protein, the total mass is \( (1.1 \times 10^{-13}) \times (1 + 2) = 3.3 \times 10^{-13}\) grams.

Step 3: Calculating Extended DNA Length

Each base pair of DNA contributes approximately 0.34 nanometers to the total length. Therefore, the extended length of the DNA is \((1 \times 10^8) \times 0.34 \times 10^{-9}\) meters, which equals 0.034 meters or 3.4 centimeters.

Step 4: Compaction Ratio Calculation

The extended length of the chromosome is 3.4 cm, which equals 34 mm. The actual length of the chromosome is given as 5 mm. The compaction ratio is \(\frac{34}{5} = 6.8\).

Step 5: Calculating Total DNA Length in Cells

Each cell contains 46 chromosomes and the extended length of DNA per chromosome is 3.4 cm. Thus, per cell, \(46 \times 3.4\) cm equals approximately 156.4 cm. For \(4 \times 10^{12}\) cells, the total is \(156.4 \times 4 \times 10^{12} = 6.256 \times 10^{14}\) cm, or \(6.256 \times 10^{12}\) meters. This is about 6.256 billion km.

Step 6: Comparing with Earth's Distance to the Sun

The distance from the Earth to the Sun is \(1.5 \times 10^8\) km. The total DNA length is about \(6.256 \times 10^{12}\) meters, or \(6.256 \times 10^9\) km. The DNA is much longer than the Earth-Sun distance.

Key Concepts

DNA mass calculation

In calculating the mass of DNA in a human chromosome, we begin by understanding the weight of a single base pair. Each base pair in DNA weighs approximately 660 daltons. To convert this weight into grams, we multiply by approximately \(1.66 \times 10^{-24}\) grams per dalton. This results in a base pair mass of about \(1.0956 \times 10^{-21}\) grams.

Given a chromosome with around \(1 \times 10^8\) base pairs, the total DNA mass of this chromosome is calculated by multiplying the number of base pairs by the mass of each base pair. The formula looks like this:

• **Total DNA Mass**: \(1 \times 10^8 \times 1.0956 \times 10^{-21} = 1.1 \times 10^{-13}\) grams

Additionally, proteins, such as histones and nonhistones, add to the chromosomal mass. If proteins account for about two grams per gram of DNA, then the chromosome's total mass, including proteins, is tripled:

• **Total Chromosome Mass**: \(1.1 \times 10^{-13} \times 3 = 3.3 \times 10^{-13}\) grams

DNA length calculation

The extended length of DNA is another fascinating aspect. Each base pair adds roughly 0.34 nanometers to the DNA strand. To find the total length of DNA in a typical human chromosome with \(1 \times 10^8\) base pairs, multiply the number of base pairs by the length per pair, as follows:

• **Total DNA Length**: \(1 \times 10^8 \times 0.34 \times 10^{-9}\) meters

You'll find the total extended length is about 3.4 centimeters (0.034 meters). When visualizing this, it's helpful to remember that if this DNA were stretched out in a linear fashion, it would reach a surprising length, indicative of the superior packaging and efficiency used by living organisms to fit DNA into a small cellular nucleus.

Chromosome compaction ratio

A crucial process in cell biology is the compaction of DNA, which ensures it fits within the confines of the cell's nucleus. Chromosome compaction ratio is a measure of how much the DNA strands are compacted. If the DNA's extended length is 34 millimeters (3.4 centimeters) and the actual length of the chromosome is only 5 millimeters, you can calculate the compaction using the following formula:

• **Compaction Ratio**: \(\frac{34 \text{ mm}}{5 \text{ mm}} = 6.8\)

This ratio of approximately 6.8 signifies that the chromosome compacts the DNA nearly sevenfold, highlighting the biological processes of winding and folding that efficiently use available space while protecting DNA from damage.

Cellular DNA content

Understanding the cellular DNA content involves not just looking at a single chromosome, but considering the entire genome replicated across all cells. Humans have around 46 chromosomes per cell. Multiplying the extended length of 3.4 centimeters per chromosome by the number of chromosomes and by the total number of cells (\(4 \times 10^{12}\)), we find the collective DNA length:

• **DNA per Cell**: \(46 \times 3.4 \text{ cm} = 156.4 \text{ cm}\)
• **Total DNA Length**: \(156.4 \times 4 \times 10^{12} = 6.256 \times 10^{14}\text{ cm}\)

Converted into kilometers, this equates to about 6.256 billion kilometers. To put it into perspective, this length exceeds the distance from the Earth to the Sun, which is about 150 million kilometers. This awe-inspiring comparison illustrates just how extensive our DNA is when considered on a global, cellular scale.