Problem 11 - Biochemistry: Concepts and Connections

Question

Assume that the translational error frequency, \(\delta\), is \(1 \times 10^{-4}\). (a) Calculate the probability of making a perfect protein of 100 residues. (b) Repeat for a 1000 -residue protein.

Step-by-Step Solution

Answer

(a) 0.9900; (b) 0.9048.

Step 1: Define the Concept

The translational error frequency \( \delta \) is the probability of an error occurring at each residue. Therefore, \( 1 - \delta \) is the probability that no error occurs (a perfect residue). When synthesizing a protein of multiple residues, each residue must be error-free for the entire protein to be perfect.

Step 2: Calculate Probability for a Single Residue

For each residue to be perfect, there must be no translational error. Thus, the probability of a single residue being correct is \( 1 - \delta \). Since \( \delta = 1 \times 10^{-4} \), the probability becomes \( 1 - 1 \times 10^{-4} = 0.9999 \).

Step 3: Calculate Probability for 100-residue Protein

The probability of synthesizing a perfect protein made of 100 residues is the probability of having 100 perfect residues in sequence. This is calculated by raising the probability of a single residue being error-free to the power of 100, thus \( (0.9999)^{100} \). Calculating gives \( (0.9999)^{100} \approx 0.9900 \).

Step 4: Calculate Probability for 1000-residue Protein

Similarly, the probability of getting a 1000-residue perfect protein is \( (0.9999)^{1000} \). Calculating this gives \( (0.9999)^{1000} \approx 0.9048 \).

Key Concepts

protein synthesis

Protein synthesis is the process by which cells create new proteins, which are crucial for various cellular functions. This multistep process involves two main stages: transcription and translation.

• Transcription occurs in the nucleus, where the DNA sequence of a gene is copied into messenger RNA (mRNA).
• Translation happens in the cytoplasm, where ribosomes read the mRNA transcript to assemble the protein.
  • The ribosome reads the mRNA in sets of three nucleotides (codons), each representing an amino acid.
  • Transfer RNA (tRNA) molecules bring the corresponding amino acids to form the growing polypeptide chain.

Errors in translation can lead to incorrect amino acids being inserted into the protein sequence, potentially affecting protein function. The frequency of such errors, called translational error frequency, can impact the overall quality of protein synthesis.

probability calculation

Probability calculations are essential to predict the accuracy of biological processes, such as protein synthesis. In the given exercise, we calculate the likelihood of synthesizing a protein without errors.

The translational error frequency, represented by \delta, is the probability of an error occurring at each step of protein synthesis. Given \delta = 1 \times 10^{-4}, the probability of a single residue being correct is calculated as follows:

• For a perfect residue, use the formula: \( 1 - \delta \).
• Substitute the value of \delta: \( 1 - 1 \times 10^{-4} = 0.9999 \).

To find the probability of forming an error-free protein:

• For a 100-residue protein: Raise this probability to the power of 100: \( (0.9999)^{100} \approx 0.9900 \).
• For a 1000-residue protein: Raise it to the power of 1000: \( (0.9999)^{1000} \approx 0.9048 \).

These calculations illustrate how even minor errors, when multiplied across many residues, affect the overall likelihood of precise protein synthesis.

amino acid sequence

An amino acid sequence is the precise order of amino acids linked together in a protein, determined by the sequence of nucleotides in the mRNA.

• Amino acids are the building blocks of proteins, and the sequence they form dictates the protein's structure and function.
• The sequence is highly specific; even a single amino acid substitution can significantly affect protein activity.

In protein synthesis:

• Each triplet of nucleotides on the mRNA (a codon) codes for a specific amino acid.
• The translation process must be accurate to ensure that the correct amino acid chain, or polypeptide, is formed.

Understanding the composition of an amino acid sequence is essential for grasping how genetic information is translated into functional proteins. Therefore, minimizing translational errors is crucial to maintaining the integrity and functionality of the proteins required by the body.