Question

Reflect and Apply Common proteins are polymers of 20 different amino acids. How big a protein (how many amino acid residues) would be necessary to have an Avogadro's number of possible sequences?

Short answer

A protein needs at least 19 amino acid residues.

Step-by-step solution

- Understand Avogadro's Number

Avogadro's number is approximately \(6.022 \times 10^{23}\). This is the number of particles (such as atoms or molecules) in one mole of a substance.

- Understand Protein Sequences

Proteins are made up of sequences of amino acids. Since there are 20 amino acids, each position in a protein sequence can be occupied by any of these 20 amino acids.

- Determine the Total Number of Sequences

If a protein consists of \(n\) amino acid residues, there are \(20^n\) possible sequences. We need \(20^n\) to be equal to or greater than Avogadro's number.

- Set Up the Equation

We want to find \(n\) such that \[ 20^n \geq 6.022 \times 10^{23} \]

- Solve for \(n\)

To solve this equation, take the logarithm of both sides: \[ \log_{10}(20^n) \geq \log_{10}(6.022 \times 10^{23}) \] \[ n \log_{10}(20) \geq \log_{10}(6.022) + 23 \] \[ n \times 1.3010 \geq 0.779 + 23 \] \[ n \times 1.3010 \geq 23.779 \] \[ n \geq \frac{23.779}{1.3010} \approx 18.29 \]

- Conclusion

Since \(n\) must be an integer, round 18.29 up to 19. Therefore, a protein would need to have at least 19 amino acid residues to have at least Avogadro's number of possible sequences.

Key concepts

Protein Length Determination

Protein length determination is crucial in understanding how many amino acid residues are required for a specific number of combinations in a sequence. In this example, the goal is to find the length necessary for Avogadro's number of possible sequences.

• Proteins are made of sequences of amino acids.
• Each protein can have various lengths, often referred to as residues.
• To reach Avogadro's number, which is about \(6.022 \times 10^{23}\) possibilities, we need to calculate the minimal length.

This requires setting up and solving an equation. By using logarithms, we simplify the comparison between possible sequences and Avogadro's number.

Amino Acid Residues

Amino acid residues are the building blocks of proteins. There are 20 different amino acids, each of which can be part of a protein sequence.

• Amino acids link together to form proteins.
• Each spot in the sequence can be occupied by any of the 20 possibilities.
• The number of possible sequences increases exponentially with each additional residue.

Understanding the concept of amino acid residues helps in determining how the length of the protein affects the total number of possible sequences.

Sequence Combinations

Determining sequence combinations involves figuring out how many unique sequences can be formed with a given number of residues. For each additional amino acid in the sequence, the combinations increase significantly.

• Each position in a sequence can be filled by one of 20 amino acids.
• For a length of \(n\), the total combinations are \(20^n\).

To address the given problem, we equate the total combinations to Avogadro's number and solve for \(n\).
Taking logarithms of both sides, we simplify:
\[ \log_{10}(20^n) \geq \log_{10}(6.022 \times 10^{23}) \] \[ n \log_{10}(20) \geq \log_{10}(6.022) + 23 \] This results in \[ n \geq \frac{23.779}{1.3010} \approx 18.29 \]. Thus, rounding up, at least 19 amino acid residues are needed to achieve Avogadro's number of sequence combinations.