Question

Most proteins contain more than 100 amino acid residues. If you decided to synthesize a " 100 -mer," with 20 different amino acids available for each position, how many different molecules could you make?

Short answer

There are \( 20^{100} \) possible different molecules.

Step-by-step solution

Understand the problem

A '100-mer' protein has 100 positions where each position can be occupied by one of 20 different amino acids. The task is to find the total number of different possible molecules.

Determine the formula

To find the total number of possible combinations, use the formula for permutations where repetition is allowed. This is calculated as the number of choices (20 for each position) raised to the power of the number of positions (100).

Apply the formula

The total number of different molecules is given by: \[ 20^{100} \]

Calculate the result

Compute the value of \( 20^{100} \) using a calculator or computational tool. The number is extremely large and typically written in scientific notation.

Key concepts

Amino Acid Sequences

Proteins are biological molecules that play many crucial roles in cells. They are made up of smaller units called amino acids. There are 20 different amino acids that can be sequenced in various ways to create a protein. This sequence, or order of amino acids, decides the protein's unique properties and function.

• Each position in a protein can be filled by any of the 20 amino acids.
• Even a small change in the sequence can affect the protein's function.

For example, one protein might have an amino acid sequence like Glycine-Alanine-Serine, while another could be Glycine-Serine-Alanine. Though similar, these two proteins may function differently in the body.
Understanding amino acid sequences is essential for grasping how proteins are formed and why each protein is unique.

Protein Structure

Protein structure is determined by the sequence of amino acids encoded in our DNA. There are four levels of protein structure:

• Primary structure: The linear sequence of amino acids.
• Secondary structure: Local folded structures that form within a polypeptide due to interactions between atoms of the backbone. The most common types are alpha-helices and beta-pleated sheets.
• Tertiary structure: The 3D shape of the protein, which is formed when the secondary structures combine.
• Quaternary structure: This occurs when multiple polypeptide chains come together to form a functional protein.

Each level of structure is dictated by the one before it.
Thus, a change in the amino acid sequence (primary structure) can affect the overall shape and function of the protein.

Permutations with Repetition

When constructing proteins, we often consider permutations with repetition. This concept involves arranging items (in this case, amino acids) where repetition is allowed.
For example, with 20 different amino acids and 100 positions to fill, the number of possible permutations can be calculated as:
\[ 20^{100} \]Here, we have 20 options for each of the 100 positions.
The formula \[ n^r \](where n is the number of options and r is the number of positions) gives us the total number of combinations.
This results in an astronomically large number, signifying the immense diversity of possible proteins that can be formed from even a relatively short string of amino acids. Understanding permutations with repetition is crucial for appreciating the complexity and variety of proteins.