Problem 53 - Essential Calculus Early Transcendentals

Question

The DNA molecule has the shape of a double helix (see Figure 3 on page 582 ). The radius of each helix is about 10 angstroms \(\left(1 \mathrm{A}=10^{-8} \mathrm{cm}\right)\) . Each helix rises about 34 A during each complete turn, and there are about \(2.9 \times 10^{8}\) complete turns. Estimate the length of each helix.

Step-by-Step Solution

Answer

The length of each helix is approximately \( 1.9 \times 10^{10} \) Å.

Step 1: Understanding the Problem

We need to estimate the length of each helix in a DNA molecule. Given parameters include the radius of the helix, the rise per complete turn, and the total number of turns.

Step 2: Convert Units

The rise per turn and radius are given in angstroms, with the relation that 1 Å = 10^{-8} cm. However, for purposes of estimation at the given magnitude, we will proceed with calculations in angstroms since conversion to centimeters will not change relative calculations significantly.

Step 3: Calculate the Length of One Turn

The length of one complete turn of the helix can be considered as the length of a spiral, which can be approximated as a diagonal of a right triangle with base being the circumference of the helix and height being the rise. Circumference = 2πr = 2π(10 Å) = 20π Å. Using the Pythagorean theorem, the length of one turn is \ \( \sqrt{(34)^2 + (20\pi)^2} \).

Step 4: Estimate Total Length of Helix

Since there are about \(2.9 \times 10^8\) complete turns, we multiply the length of one turn by the number of turns:\[ \text{Total Length} = 2.9 \times 10^8 \times \sqrt{(34)^2 + (20\pi)^2} \]

Key Concepts

Double Helix Structure

The DNA molecule is famous for its double helix structure, which looks like a twisted ladder. Imagine twisting two ladders around each other, and you get an idea of what a double helix looks like. This spiral shape is due to the way DNA strands wind around each other.

• Each strand in the helix is like a spiral staircase, where the steps are made of paired chemical bases.
• The double helix arrangement is crucial because it allows DNA to easily be copied and read, which is important for cell function and reproduction.
• The structure also provides a stable yet flexible backbone for the genetic code it carries.

This ingenious design helps protect the genetic material and ensure it is correctly passed on during cell division.

Pythagorean Theorem

The Pythagorean theorem is a useful tool in geometry and beyond. It applies to right-angled triangles, where it helps us calculate the length of one side if we know the lengths of the other two sides. The formula is:
\[ a^2 + b^2 = c^2 \]

• Here, \(a\) and \(b\) are the lengths of the two legs of the triangle, and \(c\) is the hypotenuse, the side opposite the right angle.
• In DNA helix length estimation, we use the theorem to find the length of one complete turn of the helix, treating it as the hypotenuse of a right triangle.
• The base of this triangle is the circumference of the helix, while the height is its rise per turn.

Using this theorem, we can figure out how far one complete spiral of the helix is on a straight line.

Angstroms to Centimeters

Understanding the conversion between units of measurement, like angstroms and centimeters, is crucial in science. An angstrom is a unit of length used primarily in fields like chemistry and biology to express atomic scales.

• One angstrom (Å) equals \(10^{-8}\) centimeters. This means that 1 cm is equal to 100,000,000 angstroms.
• Such conversions allow scientists to translate tiny molecular lengths into more understood units like centimeters or meters.
• The conversion factor is vital because while the calculation might be done in angstroms, the final answer may need to be expressed in centimeters for better readability or practical applications.

In many real-world applications, this conversion helps bridge the gap between microscopic science and our everyday scale.

Spiral Length Estimation

Estimating the length of a spiral involves geometric understanding and some approximation techniques. The DNA double helix is essentially two spirals intertwined, making direct measurement complicated.

• We can approximate each turn of the spiral using the diagonal of a right triangle.
• The base of the triangle is the helix's circumference, while the height is the rise of the spiral per turn.
• By applying the Pythagorean theorem, we can find the diagonal, which represents the length of one complete turn in the spiral form.

For the given problem, multiplying the length of one turn by the total number of turns gives us an estimation of the total length of the helix. Understanding this method allows us to tackle further complex structural calculations with ease.