Problem 59 - Calculus Early Transcendentals

Question

The DNA molecule has the shape of a double helix (see Figure 3 on page \(819 ) .\) The radius of each helix is about 10 angstroms \(\left(1 \mathrm{A}=10^{-8} \mathrm{cm}\right) .\) Each helix rises about 34 \(\mathrm{A}\) 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 98.6 cm.

Step 1: Understand the Problem

We need to determine the length of each helix in the DNA molecule. We know that each helix completes approximately \(2.9 \times 10^{8}\) turns, with each turn rising 34 angstroms. Using this information, we can calculate the total length.

Step 2: Calculate the Length per Turn

Since each turn of the helix rises by 34 angstroms, the length of one complete turn of the helix is 34 angstroms. We need this value to calculate the total length of the helix.

Step 3: Calculate the Total Length of the Helix

Using the number of complete turns, \(2.9 \times 10^{8}\), multiply by the length per turn. The total length is given by: \[ \text{Total Length} = 34 \text{ angstroms per turn} \times 2.9 \times 10^{8} \text{ turns}. \]

Step 4: Perform the Calculation

Calculate the multiplication: \[ \text{Total Length} = 34 \times 2.9 \times 10^{8} = 9.86 \times 10^{9} \text{ angstroms}. \] Convert to centimeters: \[ 9.86 \times 10^{9} \text{ angstroms} = 9.86 \times 10^{9} \times 10^{-8} \text{ cm} = 98.6 \text{ cm} \].

Key Concepts

Double Helix

The term "double helix" is used to describe the structure of DNA. Imagine a twisting ladder; that's what a double helix looks like. This shape is critical because it allows DNA to store and transmit genetic information efficiently. DNA's structure consists of two long strands that wind around each other, symbolizing a spiral staircase. These strands are made up of nucleotides which include a sugar, a phosphate group, and a nitrogenous base. The bases pair specifically, forming the steps of our staircase: adenine pairs with thymine, and cytosine pairs with guanine. Understanding the double helix is crucial as it explains how genetic information is copied and transferred. The helix formation allows for compact storage of a vast amount of information within a cell. This shape is also vital during processes like DNA replication, where each strand serves as a template to produce a new complementary strand.

DNA Molecule

DNA stands for Deoxyribonucleic Acid, which is the molecule responsible for carrying genetic instructions used in growth, development, functioning, and reproduction of all living organisms. Essentially, DNA is the blueprint for all biological entities. Composed of two strands, the DNA molecule twists into a double helix. Each strand is a long chain of nucleotides that carry genetic information through sequences of A, T, C, and G. These sequences are like a code that instructs cells on which proteins to build. DNA molecules are incredibly compact, yet they carry an immense amount of information required for the functioning of any living organism. They also have a pivotal role in heredity as they contain genes passed from parents to offspring.

Helix Length Calculation

Calculating the helix length of a DNA requires understanding the structure and how it translates into physical dimensions. In our example, each complete turn of the helix rises by a specific distance, 34 angstroms, which can be converted into centimeters if needed. The length of a DNA helix can be calculated by the number of turns and the height gained per turn. By multiplying the height of each turn by the total number of turns \[\text{Total Length} = 34 \text{ angstroms per turn} \times 2.9 \times 10^8 \text{ turns} = 9.86 \times 10^9 \text{ angstroms}\]The conversion from angstroms to centimeters helps in understanding the proportionate length of DNA in more conventional units.

Biological Mathematics

Biological mathematics bridges the fields of biology and mathematics to help explain life's complexities using mathematical models and equations. This discipline is crucial for solving real-life problems, such as calculating DNA structures or understanding population dynamics. When we look at something like a DNA molecule, mathematical models help calculate lengths, angles, and various properties necessary for biological functions. Calculations such as those involved in helix length estimation are great examples of the application of mathematics in biology. Using mathematical tools to interpret biological data can help in areas like genetic engineering, epidemiology, and ecological modeling. So, understanding the basics of biological mathematics can provide deeper insights into how life operates on both micro and macro levels.