Question

Nucleic acids are polymers of just four different monomers in a linear arrangement. How many different sequences are available if one makes a polymer with only 40 monomers? How does this number compare with Avogadro's number?

Short answer

There are \( 4^{40} = 1.4615016 \times 10^{24} \) sequences, which is slightly more than twice Avogadro's number.

Step-by-step solution

- Understand the monomers

Nucleic acids are made up of four different monomers: adenine (A), thymine (T), cytosine (C), and guanine (G).

- Calculate the number of possible sequences

For a polymer with 40 monomers, each position can be one of the four monomers. Therefore, the total number of sequences is calculated as: \[ 4^{40} \]

- Calculate the exponential expression

Evaluate the expression: \[ 4^{40} = 1.4615016 \times 10^{24} \]

- Compare with Avogadro's number

Avogadro's number is approximately \[ 6.022 \times 10^{23} \]. Compare the two numbers: \[ 1.4615016 \times 10^{24} > 6.022 \times 10^{23} \] A polymer of 40 monomers has a number of different sequences that is slightly more than twice Avogadro's number.

Key concepts

Polymer

Nucleic acids like DNA and RNA are examples of **polymers**. A polymer is a large molecule made up of repeating smaller units called monomers.
Think of a polymer as a chain and the monomers as the links in that chain. In the case of nucleic acids, the chain is a long sequence of nucleotides, and each nucleotide is a monomer.
This long sequence can be varied to form different genetic codes, which are crucial for the biological functions of living organisms. For instance, DNA (a type of nucleic acid) contains the instructions needed for an organism to develop and function.

Monomers

The building blocks of nucleic acids are called **monomers**. In DNA, the four monomers are adenine (A), thymine (T), cytosine (C), and guanine (G).
Each monomer can be linked in a variety of sequences, making up the polymer. Monomers are like characters in a story. You can use them in different ways to make words, sentences, and stories (or in this case, genetic information). This flexibility allows for the immense variety of life's genetic codes.
In a polymer with 40 monomers, you can have many different combinations, making each sequence unique.

Avogadro's Number

**Avogadro's number** is a fundamental constant in chemistry that is used to count entities like atoms, molecules, or ions. It is approximately \[ 6.022 \times 10^{23} \]. This number is so large because it allows chemists to work with amounts of substances that are easier to handle in a laboratory.
To put it in perspective, if you have that many grains of rice, you could cover the Earth's surface to a depth of several meters!
When comparing this number to the number of different sequences possible for a polymer with 40 monomers, which is \[ 4^{40} = 1.4615016 \times 10^{24} \], you can see that the number of sequences is slightly more than twice Avogadro's number. This showcases the vast complexity and diversity present in nucleic acid sequences.

Exponential Expression

An **exponential expression** is a mathematical expression involving exponents, where a number is raised to the power of another number. In the context of nucleic acid sequences, we use the exponential expression \[ 4^{40} \] to calculate the total number of possible sequences for a polymer made of 40 monomers.
Here, the base 4 represents the four different monomers (A, T, C, G), and the exponent 40 represents the number of monomers in the polymer.
Evaluating \[ 4^{40} \] gives us \[ 1.4615016 \times 10^{24} \], a vast number illustrating the immense potential diversity of nucleic acid sequences. This exponential growth in the number of possible sequences is fundamental in understanding the complexity of genetic information.